Analytical and numerical methods for wave propagation in fluid media

Analytical and Numerical Methods for Wave Propagation in Fluid Media

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS

Founder and Editor: Ardeshir Guran Co-Editors: A. Belyaev, H. Bremer, C. Christov, G. Stavroulakis & W. B. Zimmerman

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Vol. 7

Analytical and Numerical Methods for Wave Propagation in Fluid Media Author: K. Murawski

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series A

Volume?

Founder and Editor: Ardeshir Gliran Co-Editors: A. Belyaev, H. Bremer, C. Christov, G. Stavroulakis & W. B. Zimmerman

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in Uiiiil Miedifi

K. Murawski Uniwersytet Marii Curie-Skfodowskiej, Lublin, Poland

1111% World Scientific

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This book is dedicated to my wife, Ewa, and son, Kamil, for their love and support.

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Preface

Much of the matter filling the Universe is in a state of fluid which comprises liquid, gas, and plasma. The mathematical description of fluid motion makes use of partial differential equations which propagate the fluid variables in time. Many phenomena that occur in the fluid, treated as a continuous medium, can be studied in the frame of hydrodynamics (HD) or magnetohydrodynamics (MHD), which is a relevant and simple tool to describe the behaviour of the fluid. Among many extraordinary phenomena found in fluid such as, for instance, convective cells (see the Figure on the book cover for the convective cell seen in the Bieszczady Mountains in Poland on June 22, 2000), waves are the most pervasive and can be observed everywhere in the Universe (molecular clouds, extragalactic jets, accretion disks, Sun, solar wind, magnetospheres, magnetotail, cometary tails). We will point out the ubiquity of wave phenomena in fluids by discussing few selectively chosen examples such as acoustic waves, emission of air-pollutions, magnetohydrodynamic waves in the solar corona, solar wind interaction with Venus, and ion-acoustic waves. This book is primarily concerned with the analytical and numerical solutions to hyperbolic system of wave equations. Mathematically the most interesting feature of such systems is that they admit a shock solution which is a discontinuity that can form as a consequence of nonlinearity even from small initial condition after sufficiently long time. No attempt has been made in this book to study in any detail the physics of fluids and waves, or to explore the physical significance of the problems we solve using the presented analytical and computational techniques. Although the reader is expected to be conversant in the basic concepts of fluid

vu

Vlll

Preface

dynamics, as expounded, for example, in Landau and Lifshitz (1986) this book consists an overview of main ideas and approaches with a hope that they will serve as a roadmap to students or researches new to this area. This book is not intended to be complete. Some of the results are presented without derivation and it is left to the reader to consult the references provided for a complete presentation. Very often a considerable effort is required to add the details that have been omitted. However, our intention is that the reader will find this overview a useful introduction to the analytical and numerical methods for solving nonlinear equations which describe wave propagation in fluid media. While this book is in no sense a treatise on the whole subject of analytical and numerical techniques for fluid dynamics, it spans a relatively wide range of topics. The emphasis is on methods in use by the author, although brief descriptions of other techniques are included as background material too. We describe those techniques which are of greatest performance and are most widely used. Likewise, the references are for the most part to most-known papers but there are many approaches and interesting works which are not cited or surveyed here. This book would never appear without my teachers and coworkers. I wish to express my profound gratitude to all of them but in particular to Profs. Wieslaw A. Kamiriski and Eryk Infeld for their unfailing support and the latitude they gave me in choosing my sometimes meandering research direction, and to Prof. Bernard Roberts for his introduction to solar physics. Thanks are also due to many coworkers (Jose Luis Ballester, Rick DeVore, Sasha Kosovichev, Randy LeVeque, Valery Nakariakov, Luigi Nocera, Ramon Oliver, Efim Pelinovsky, Tomasz Plewa, Mike Ruderman, Rich Steinolfson, Oskar Steiner, Robin Storer, Takashi Tanaka, Rolf Walder, Ivan Zhelyazkov, and others) who have proven to be excellent guides through the jungle of problems some of which are undertaken in this book. Notification of errors, criticism, and suggestions for improvement are welcome.

K. Murawski Lublin 2002

Contents

Preface

vii

Chapter 1 Introduction 1.1 Limitations of analytical and experimental methods 1.2 Numerical simulations — a bridge between analysis and experiment 1.3 Advantages of computer simulations 1.4 Scope of the book Chapter 2 Mathematical description of 2.1 Classification of differential equations 2.1.1 Advection equation 2.1.2 Characteristics of the advection equation 2.2 A conservation equation 2.3 The Navier-Stokes equations 2.4 The one-dimensional Euler equations 2.5 Plasma and the Maxwell's equations 2.6 Kinetic plasma 2.7 Quasi-particle approximation 2.8 Magnetohydrodynamic approximation 2.8.1 Lagrangian picture 2.8.2 Incompressible limit 2.8.3 Cold plasma limit Chapter 3 Linear waves 3.1 Waves in homogeneous fluids

fluids

1 1 2 3 4 7 7 8 9 10 12 13 14 15 18 19 22 22 23 25 26

ix

x

Contents

3.1.1

3.2

MHD waves 3.1.1.1 The Alfven wave 3.1.1.2 Magnetosonic waves 3.1.2 Ion-acoustic waves Waves in inhomogeneous fluids 3.2.1 Acoustic and internal gravity waves in gravitationally stratified medium 3.2.2 Sound waves in random fields 3.2.2.1 Waves in random mass density field

Chapter 4 Model equations for weakly nonlinear waves 4.1 Inviscid Burgers equations for fast MHD waves 4.1.1 Nonlinear interactions 4.1.2 Modified inviscid Burgers equation 4.1.3 Inviscid Burgers equation 4.2 The Burgers equation for acoustic waves in viscous fluid . . . . 4.3 The Korteweg-de Vries equation for long waves in a cylinder . . 4.4 Few modifications of the KdV equation 4.5 The Zakharov-Kuznetsov equation for strongly magnetized ionacoustic waves 4.6 The nonlinear Schrodinger equation for modulational waves in a cylinder 4.6.1 Few remarks on the NS equation 4.7 Few other model wave equations 4.8 Remarks on multi-dimensional wave equations

33 36 37 45 45 46 47 49 50 52 56 58 58 60 61 62

Chapter 5 5.1 5.2 5.3 5.4

5.5

Analytical methods for solving the classical model wave equations Analytical solution of the inviscid Burgers equation The direct method for the Burgers equation Backlund transformation for the Korteweg-de Vries equation . 5.3.1 Solitons Inverse scattering method 5.4.1 Lax criterion 5.4.2 Inverse method Stationary wave solutions of the nonlinear Schrodinger equation

26 28 29 30 33

65 65 66 69 70 71 71 72 74

Contents

Numerical methods for a scalar hyperbolic equations 6.1 Finite-difference approximations 6.2 Simple finite-difference schemes 6.2.1 Lax-Wendroff and Lax-Fredrichs schemes 6.2.2 The Beam-Warming method 6.2.3 The MacCormack scheme 6.2.4 A stable scheme 6.2.5 Hermitian compact scheme 6.2.6 Upwind differencing 6.2.7 Method of lines discretization 6.3 Temporal discretization 6.3.1 Runge-Kutta methods 6.3.2 Multigrid methods 6.3.3 Linear multi-step methods 6.4 Finite-volume methods 6.5 Von Neumann stability analysis 6.6 Explicit and implicit time integrations 6.6.1 An implicit scheme 6.6.2 Semi-implicit method 6.7 Numerical errors 6.7.1 Spurious modes 6.7.2 Overshoots and undershoots 6.7.3 Monotonicity, positivity, and causality 6.8 Problems with source terms 6.8.1 A fractional step method 6.9 Open boundary conditions 6.10 Shock-capturing schemes 6.10.1 Algebraic schemes 6.10.2 Geometric schemes 6.10.3 Godunov method 6.10.4 Riemann problem 6.10.5 The MUSCL scheme 6.10.6 Higher-order schemes 6.10.7 Kurganov schemes 6.11 Flux-corrected transport method 6.11.1 Convection 6.11.2 Diffusion 6.11.3 Anti-diffusion

xi

Chapter 6

79 79 81 83 84 85 85 85 86 88 89 89 90 90 91 93 94 95 95 97 100 100 101 104 104 105 107 107 108 108 108 109 Ill 112 113 114 115 115

xii

Contents

6.12 Spectral methods 6.12.1 The Fourier transform method 6.12.2 The Chebyshev expansion method 6.13 Finite-element method 6.14 The locally one-dimensional method

Chapter 7

Review of numerical methods for model wave equations

116 118 119 121 122

123

Chapter 8

Numerical schemes for a system of one-dimensional hyperbolic equations 127 8.1 Linear system of one-dimensional equations 127 8.1.1 Characteristic variables 128 8.1.2 Riemann problem for the linear equations 129 8.1.3 The wave propagation method 130 8.2 Nonlinear system of one-dimensional equations 131 8.2.1 Flux-difference splitting scheme 131 8.2.2 Euler equations 133 8.3 The shock tube problem 135 8.4 Rankine-Hugoniot jump condition 136 8.5 The Riemann problem for the Euler equations 137 8.5.1 The HLL Riemann solver 139 8.5.2 The Roe approximate Riemann solver 139 8.5.3 A relaxation Riemann solver 141 8.5.4 Extension of the Roe scheme for a general equation of state 142 8.6 Deficiences of Godunov-type schemes 144 8.6.1 Entropy fix 144 Chapter 9

A hyberbolic system of two-dimensional equations 9.1 Operator splitting schemes 9.2 Operator unsplit methods 9.3 Grid generation 9.3.1 Structured and unstructured grids 9.3.2 Other grid generation methods

149 150 151 152 152 154

Contents

9.4 9.5 9.6

Adaptive mesh refinement method 9.4.1 AMR codes Implicit hydrodynamic schemes 9.5.1 Barely implicit scheme for the Euler equations Few specific examples of hydrodynamic schemes

xiii

. . . .

156 157 158 161 163

Chapter 10 Numerical methods for the M H D equations 165 10.1 Problems with the MHD equations 166 10.2 Conservative form of the MHD equations 167 10.3 Non-conservative equations 168 10.4 Eigenvalues and eigenvectors 170 10.5 Singularities 171 10.6 Problems with MHD Riemann solver 173 10.7 Divergence cleaning schemes 173 10.8 A scheme for a strong magnetic field 176 10.9 Few specific examples of explicit MHD schemes 178 10.9.1 9-th wave Riemann solver for two-component MHD equations 179 10.10 Implicit MHD schemes 184 Chapter 11 Numerical experiments 11.1 Numerical solution of the inviscid Burgers equations 11.2 The effect of random mass density fields on acoustic waves . . 11.2.1 Seeding time-dependent random field 11.2.2 Numerical results 11.2.3 Summary 11.3 Numerical simulations of air-pollutions 11.3.1 Numerical model 11.3.2 Numerical results and discussion 11.4 Driven MHD waves in the solar corona 11.4.1 Physical model 11.4.2 Numerical solution of MHD equations for the solar coronal plasma 11.4.3 Numerical results 11.4.4 Summary 11.5 Solar wind interaction with Venus 11.5.1 Numerical model

187 188 191 192 193 194 194 195 196 199 200 200 201 208 209 212

xiv

Contents

11.5.2 Numerical results and discussion 11.5.3 Concluding remarks 11.6 Ion-acoustic waves and solitary waves

214 216 217

Chapter 12

221

Summary of the book

Bibliography

223

Index

235

Chapter 1

Introduction

1.1

Limitations of analytical and experimental methods

Traditionally the scientific methods involve a mutual interplay between experiment and analysis. The former tries to collect information by repeated events. The latter attempts to order the accumulated knowledge. Analysis and experiment interact with each other via mutual stimulation and feed back. However, the traditional methods of investigating nature have their limitations. Often the complexity of the physical phenomenon and the simultaneous interaction of various effects make a complete analysis impossible. On the experimental side, one is limited to measurements of only a small fraction of the quantities of interest and even they can be sampled only at a few times and spatial locations and with a limited degree of accuracy. Consequently, one is then faced with the task of interpreting limited observations with theories that are incomplete. For example for astrophysical plasma, it is usually impossible to repeat experiments and the available observational data is sparse and sporadic. The observational data is mainly collected by satellite observations, space active experiments and ground based active or passive remote diagnostics. Consequently, the accumulation of such data is an extremely expensive and time consuming task. Additionally, the data very often depends on time and space. The sheer volume of exploration in space is so large that the essentially point observations by spacecraft alone often leads to misinterpretation of the data. Computer simulations can reproduce the global phenomena which can never be measured simultaneously even by a number of spacecrafts. The present computers can be used to model a macroscopically neu-

1

2

Introduction

tral medium containing free electrons and ions. This medium constitutes a plasma when the long-range Coulomb forces produce collective behavior. The need for time-dependent calculations of plasma is stimulated by observations and theory which show the existence of time-dependent behavior. The development of supercomputers makes detailed and accurate calculations with relatively short calculation times feasible.

1.2

Numerical simulations — a bridge between analysis and experiment

Numerical simulations - that is, the use of computers to solve problems by simulating theoretical models - is part of new methodology that has taken its place alongside pure theory and experiment during the last few decades. Numerical simulations permit one to solve problems that may be inaccessible to direct experimental study or too complex for theoretical analysis. Computer simulations can bridge the gap between analysis and experiment. Very often the simplest physical phenomena are described by complicated mathematical equations which cannot be solved analytically and require numerical treatment. The basic idea of computer experiments is to simulate the physical behavior of complicated natural systems by solving an appropriate set of mathematical equations that are built on the basis of a physical model. A typical way for computer simulation is to develop a mathematical model, perhaps in a series of differential or integral equations and then to transform them to a discrete form that can be numerically treated. By this way, numerical simulations attempt to initiate the dynamic behavior of a system and to predict or calculate subsequent events. Numerical simulations have emerged as a new branch in physics complementing both experiments and theory. A simulation can sometimes replace a physical experiment, although most often a simulation and an experiment are complementary. Results of scientific experiments are often explained by simulations, and simulations are often calibrated by experiments. The experiments provide input for the simulations which are viewed as experimenting with theoretical models. The feedback of numerical results into theoretical modeling and the continues interaction with laboratory experiments and analytical theory make computing an indispensable tool for science. Therefore, the increase in computing power in both speed and

Advantages

of computer

simulations

3

storage has given computational physics its significance. Improved computer capacity and the solution algorithms themselves, have a large effect on the quality of solutions obtained. Numerical simulations can be used to study the dynamics of complex physical systems. Although the variety of complex flows that computational fluid dynamics can analyze continues to increase, the solutions to much more complex flows are desired. A numerical model can be used to interpret measurements and observations, extend existing analytical models into new parameter regimes and quantitatively test existing theories. That can be done by comparing model predictions to experimental data. Modern computers are fast and do not complain of boredom when repeating the same procedure millions of times. Analytical methods, on the other hand, have been plagued with this problem. With the use of computer, one can often test theoretical predictions and approximations. The numerical models are simpler and more idealized than the actual physical system. However, they are far more complete and realistic than we can handle analytically.

1.3

Advantages of computer simulations

Computer simulations contain many advantages over conventional experiments. Simulations can evaluate the importance of a physical effect by turning this effect on or off, changing its strength, or changing its functional form. This way of isolating effects is an important advantage that a simulation has over an experiment. The main advantage of computer experiments is that complicated physical system involving nonlinearity and inhomogeneity can be treated without difficulty as easily as much simpler linear and homogeneous systems are dealt with. As a consequence of that, nonlinearity and inhomogeneity is no longer an obstacle in exploitation of physical systems. The computer simulations reproduce both linear and nonlinear behavior of a physical system. One can compare the results of such calculations with the behavior of real physical systems and with theory. These results can then be used to test theoretical predictions. Both laboratory measurements and computer simulations are never exact. Theoretical researcher, by contrast, is blessed with the possibility of generating exact solutions. The computer does not treat the analytical for-

4

Introduction

mulae in a way a theoretical physicist is fond of manipulating. Instead, many bits of numbers are crunched by the computer. Consequently, one can get only a single event instead of a physical law out of the computer. To learn the general behavior and laws of nature, we have to interpret and analyze the computer results. Both simulations and laboratory experiments benefit greatly from focusing on specific mechanisms of complex phenomena. Therefore, much can be learned about physical phenomena by idealizing and simplifying the problem. As a consequence of that, an experiment is not always a better probe of a physical system than a simulation. Simulations can be used to test the range of validity of theoretical approximations. For example, when a linear theory breaks down, the reason of breakdown can be studied by simulations. So, the simulations can be used to test and extend analytical results. The reverse case is also true as theory can be used to validate a numerical model. Numerical simulations, analysis and experiment cover mutual weakness of both pure experiment and pure theory. These simulations will remain a third dimension in fluid dynamics, of equal status and importance to experiment and analysis. It has taken a permanent place in all aspects of fluid dynamics, from basic research to engineering design. The computer experiment is a new and potentially powerful tool. By combining conventional theory, experiment and computer simulation, one can discover new and unsolved aspects of natural processes. These aspects could often neither have been understood nor revealed by analysis or experiments alone.

1.4

Scope of the book

The purpose of this book is to present few analytical and numerical methods of solving wave propagation problems. The book is organized along the following guidelines. Chapter 2 is devoted to mathematical descriptions of fluids. Here several types of plasma approximations are presented. They include equations for hydrodynamics and magnetohydrodynamics. In Chapter 3 the linear dispersion relations for waves in homogeneous and inhomogeneous media are reviewed with a description of the problems which appear when inhomogeneous media, such as stratified by gravity or contained random fields, are taken into account. The following Chapter presents several model equations

Scope of the book

5

for weakly nonlinear waves and solitons. A few analytical methods of solving these equations are described in Chapter 5. Chapters 6-10 introduce various numerical methods, pointing out their strengths and weaknesses. Some of these methods are used to solve the inviscid Burgers equations and in the simulations of random acoustic waves, air-pollutions, waves in the solar corona, solar wind interaction with non-magnetic bodies such as Venus, and ion-acoustic waves. These numerical experiments are performed in Chapter 11. This book is completed by Summary and the list of references as well as by the subject index.

Chapter 2

Mathematical description of fluids

2.1

Classification of differential equations

We consider the following second-order differential equation: autXX + butXt + cu,tt + dutX + eu,t + gu = s,

(2.1)

where s is a source term, u = u(x, t) and the comma with indices denote the partial differentiation, e. g. U xx

'

_d2u ~ Ox2'

This notation for partial derivatives will be used throughout this book. Classification of differential Eq. (2.1) is based on the sign of the discriminant A = b2 - 4ac.

(2.2)

For A < 0, A = 0, and A > 0 we have respectively the elliptic, parabolic, and hyperbolic equations. These equations model different sorts of physical phenomena in fluid dynamics and require different analytical and numerical methods for their solution. Typical examples are Poisson equation, u,xx + u,tt = s,

(2.3)

for an elliptic equation, the diffusion equation, u,t = (£>«,X),B, 7

(2.4)

8

Mathematical

description

of fluids

for a parabolic equation, and the wave equation, ujt ~ c2utXX = 0,

(2.5)

for a hyperbolic equation. Here D{x) is the diffusion coefficient and c is the wave speed. For c = const this equation can be rewritten as a set of two first-order equations vtt + cw,x

=

0,

(2.6)

w,t + cv,x

=

0,

(2.7)

where: v = — cuiX,

w = utf

(2.8)

For s = 0, Eq. (2.3) is called Laplace'a equation. When a, b, c, d, e, g or s are functions of u or space and time, Eq. (2.1) is nonlinear and the class of an equation may vary according to the sign of A in particular time and location in space. 2.1.1

Advection

equation

This book is primarily concerned with first-order hyperbolic system of n equations of the following form: u t + Au

x

+ Bu,y + C u

z

= 0,

(2.9)

where u(x, t) 6 Rn and A, B, C are nxn matrices which have real eigenvalues, and are diagonalizable, i. e., have complete sets of linearly independent eigenvectors. Equation (2.9) is equivalent to Eqs. (2.6) and (2.7) if

and B = C = 0

(2.11)

or d/dy = d/dz = 0. In fluid dynamics u represents conserved quantities such as the densities of mass, momentum, and energy.

Classification

of differential

9

equations

The simplest example of a hyperbolic equation is the scalar advection equation uit + cu,x=0

(2.12)

for which the initial condition is given by u(x,0)=uo(x).

(2.13)

Here, c = const can be thought as a convective velocity for the generalized density u. Equation (2.12) can be easily solved by rewriting it as

i + «5)- a

<2 14)

-

So, the solution of the above Cauchy (initial-value) problem can be expressed as u(x,t)=u0(x-ct,0).

(2.15)

That is, the initial profile is simply translated with a vector ct without any change in shape. Generally speaking, c in Eq. (2.12) can be a function of x, t, and u, leading to a nonlinear advection equation, u,t + f(u),x

= 0,

(2.16)

where f(u) is a flux. Its quasi-linear analog can be written as follows: u,t + /,„«,« = 0.

(2.17)

It is noteworthy here that while wave Eq. (2.5) describes bi-directional waves, advection Eq. (2.12) corresponds to a uni-directional wave which propagates to the left or right, depending on the sign of the convective velocity c. 2.1.2

Characteristics

of the advection

equation

The characteristics of the nonlinear advection Eq. (2.17) are curves in the x — t plane such that dx — = /,„,

x(t = 0) = x0.

(2.18)

10

Mathematical description of fluids

For the advection Eq. (2.12) this expression becomes ^ = c, at

x(t = 0) = x0.

(2.19)

So, the characteristics are the straight lines x = xo + ct through each point xo at time t = 0. Then, u{t) = u(x0 + ct,t)

(2.20)

and u(t)>t = cu,x+utt

= 0.

(2.21)

So, the solution u(t) is constant along each characteristic. It will be shown in Sees. 6.10 and 8.1 that characteristics are important for developing accurate numerical schemes for hyperbolic equations.

2.2

A conservation equation

Many physical processes are governed by fundamental laws such as a conservation of mass, energy, momentum, and charge. These processes can often be described by the conservation equation which is a generalization of Eq. (2.16), viz. Q,t + V • (gv) = git + v • Vg + pV • v = 0.

(2.22)

Here g is a generalized density and v is a flow speed. The term v • Vg describes convection and the expression gV • v corresponds to compression or to the reverse effect, rarefaction. We will derive now the equation for conservation of mass in a onedimensional gas dynamic system. A more general derivation can be found, for example, in Landau and Lifshitz (1986) and Wendt (1992). Let x and g(x,t) represent the distance and the mass density at the point x, respectively. If we assume that there are nor sinks nor sources (mass is neither created nor destroyed) then the mass in section < x\, x-i > can be changed only because of fluid flowing across the end-points x\ and

A conservation equation

11

xi of this section (Fig. 2.1). We can express that by the following formula: d d~t

rX2 nX2

I

g{x,t)dx = g(xi,t)v{x1,t)-Q(x2,t)v(x2,t),

(2.23)

J X\

where v(x,t) is a fluid velocity and the product g(x,t) and v(x,t) is equal to the flux of fluid. From this equation we get / J X\

Qttdx = - /

(gv)iXdx

(2.24)

JX\

or

[g,t + (gv),x]dx = o.

(2.25)

Since this equality holds for any interval < xi, x2 > the integrand must be zero, i. e. (2.26)

g,t + (QV),X = 0.

This is the one-dimensional counterpart of the differential form of the mass continuity equation. In the case of constant velocity v in Eq. (2.26) we obtain the simple advection equation, (2.27)

g,t + vgyX = 0.

*>

X

X

Fig. 2.1 The mass is changed in the interval < xi, X2 > as a result of inflow from the left and outflow at x = X2Equation (2.26) can also be written in Lagrangian form as: dg g,t + vg,x = -£=

~ev,x,

(2.28)

where the two terms on the left hand side are the Eulerian time derivative and the advective space derivative. Together these two terms comprise the

Mathematical description of fluids

12

Lagrangian derivative dg/dt which denotes the change of Q in a reference frame, moving with the speed v.

2.3

The Navier-Stokes equations

The Navier-Stokes equations were obtained in the first half of the nineteenth century (in 1845) independently by the Frenchman M. Navier and the Englishman G. Stokes. In Cartesian coordinates, the Navier-Stokes equations for two-dimensional flows (with d/dz = 0) can be written as the set of conservation equations of Eq. (2.22), viz. (

\

\

QVX T

QVX

QK ~ xx

+

QVy

QVxVy

V EJ

\

+

7~xy

(E + p)vx - VXTXX - vyTxy + qx J

\

QVy

V

Qvl - r, y -vv (E+p)vy- VXTyX

. VyTyy

~\~ Qy

Here S is the source term, the velocity v = [vx,vy,0], energy density such that the pressure p is given by, P = (7 ~ 1)

= s.

(2.29)

/

and E is the total

E-jQtvl+tf)

(2.30)

The specific heats ratio 7 = cp/cv is such that 7 = (m + 2)/m, where m is the number of internal degrees of freedom of the fluid molecules, cp and cy are the specific heats at constant pressure and volume, respectively. For a monoatomic fluid m — 3, while for diatomic molecules m = 5. In the late seventeenth century Isaac Newton claimed that shear-stress T in a fluid is proportional to velocity gradients. Such fluids are called Newtonian fluids in opposite to non-Newtonian fluids such as blood for which this dependence does not exist. For Newtonian fluids Stokes showed that the normal (TXX, ryy) and shear (TXV, ryx) stresses are TXx

-

T~xy

=

AV -v + 2fivXtX, ^\Vx,y

T Vy x),

Tyy = W -y + 2fxvyy, TyX =

TXy,

(2.31)

The one-dimensional

Euler

equations

13

where fj, is the molecular viscosity coefficient and A is the bulk viscosity coefficient. The normal stresses are related to the time rate of change of volume of the fluid element, whereas the shear stresses are associated with the time rate of change of the shearing deformation of the fluid element. Stokes made the hypothesis that

This relation is frequently used but is not definitely confirmed. For gases, the viscosity coefficient fj, is assumed to vary in accordance with Sutherland's law (Quirk 1991), / T \ 3 / 2 7 o + 110

^°UJ

r+iio'

(2 32)

"

where free stream conditions are denoted by the subscript 0. Heat transfer by thermal conduction is proportional to the local temperature gradient qx —

qy =

-KT,X,

~KT
(2.33)

where K is the thermal conductivity coefficient. 2.4

The one-dimensional Euler equations

The Euler equations are derived from the Navier-Stokes Eqs. (2.29) by setting the viscosity \x and thermal conduction K to zero. In the onedimensional case (d/dy = d/dz = 0) Euler equations can be written in explicitly Q,t + (Qv),x = S„, 2

(<*>),* +(0W +P),* = O, Ett + [v(E+p)],x

= 0,

(2.34) (2.35) (2.36)

where E = p/(j—l)+gv2/2 is the total energy density, v is the x-component of velocity, p is the pressure, and Se represents the external mass flux which corresponds to sinks (Se < 0) or sources (Se > 0). The other sources in these equations are neglected. Note that while the mass density g and the momentum gv are convected with the speed v, the effective speed at which energy is transported is veff =

14

Mathematical

description

of fluids

v(E + p)/E. This fact has direct consequences in developing numerical methods for the Euler equations.

2.5

Plasma and the Maxwell's equations

The term plasma was first coined by Tonks and Langmuir (1929) in there studies of oscillations in electric discharges. Plasma can be generated thermally. As heat is added to a solid such as for instance ice, it undergoes a phase transition to a new state which is called liquid. If heat is further added to a liquid, it becomes the gaseous state. An additional energy input results in the ionization of some of the atoms. At a sufficiently high temperature (that usually exceeds 105 K) most matter exists in an ionized state which is called plasma. A plasma state can exist at temperatures lower than 105 K if there is a mechanism for ionizing the gas and if the mass density is low enough so that recombination is not rapid. Unlike conventional gases but like most liquids, plasma is capable of conducting an electric current. Since this plasma state behaves quite differently from the other states, it is often called the fourth state of matter. This term was introduced by Crookes (1879) to describe the ionized medium generated in a gas discharge. Although there is little in the way of natural plasma on the Earth's magnetosphere as the low temperature and high density of the Earth's near atmosphere preclude the existence of plasma, it exists in a variety of places, some as familiar as a fluorescent light or perhaps less well-known locations such as the leading edge of high speed space-shuttles. Plasma exists in the upper atmosphere (ionosphere) too where it is created by photoionization of the tenuous atmosphere. Father out from the Earth, plasma streams from the Sun in the form of the solar wind which consists mainly of protons, and fills many regions of interstellar space. It is estimated that as much as 99.9% of the universe is comprised of plasma. Many phenomena in plasma can be described by the Maxwell's equations (e. g., Dendy 1990) which can be written as follows: V B

=

0,

(2.37)

V-E

=

-,

(2.38)

-VxB

=

£E,t-rj,

(2.39)

Kinetic plasma

VxE

=

-Bt.

15 (2.40)

In these equations the relations B = /iH and D = eE have been used to eliminate the magnetic field strength H and electric displacement D . Here E is the electric field, q is the charge density, fj, and e are respectively the magnetic permeability and permittivity. For vacuum these values are H = 4?r • 1(T 7 H/m and e ~ 8.854 • 1CT12 F/m. The symbol j denotes the current density that is given by Ohm's law j = a(E + v x B ) ,

(2.41)

where a is the electric conductivity. The Maxwell's equations have a simple physical interpretation. Equation (2.37) implies that there are no magnetic monopoles, whereas Eq. (2.38) and Faraday law (2.40) assume that either electric charges or timevarying magnetic fields are able to generate electric fields. Ampere law (2.39) shows that either currents or time-varying electric fields can give rise to magnetic field.

2.6

Kinetic p l a s m a

There are situations in geophysics (e. g., Shizgal, Hubert 1989) and in space physics (Shizgal, Blackmore 1986) where the mean free path of particles, which is the average distance traveled between particle collisions, is comparable to, or higher, that a local spatial scale. For these rarefied gas dynamical situations the Euler equations are no longer valid and a kinetic theory treatment is required. We consider i-th species of the plasma, represented by particles with electric charge e*, mass rrii, number density rij, velocity v, and temperature T{. Kinetic processes are characterized by time-scales which are associated with the plasma frequency,

«>i = J^-,

(2-42)

and the cyclotron frequency in the magnetic field B,

fl, = ! * £ . rn.i

(2.43)

16

Mathematical description of fluids

Length-scales involve these frequencies and relevant velocities. They are: a) the Debye length,

XDi = J ^ 4 t

(2-44)

with the thermal speed

c.=

/MI V mi

Here ks is Boltzmann's constant; b) the thermal gyroradius

m=

Ci

(2.45)

c) the direct gyroradius rdi

=

Ai

=

Vi

(2.46)

d) the inertial length

c

(2.47)

A kinetic (fluid) plasma description is required when these scales are higher than or comparable to (smaller than) the scales of physical phenomena. There is also an intermediate regime, for example, when kinetic effects along an ambient magnetic field are important but they are negligible in the perpendicular direction. Alternatively, the various plasma species can be treated differently. In a hybrid approach one species is treated kinetically and the other species are considered as a fluid. The gaseous state can be described by the distribution function for the different species. For sufficiently rarefied gases, the single distribution function, / ( v , r, t), is sufficient for this purpose, assuming that there are no two particle correlations. The distribution function is defined as the quantity f(v,r,t)dvdr is equal to number of particles with velocity in the interval < v, v + d v > and position < r, r+dr > at time t. This distribution function depends on seven dependents: three components of velocity, three position variables and the time. A reduction of the number of dependents is required to make the problem tractable. This reduction is achieved by expressing the distribution function in terms of moments.

Kinetic plasma

17

The first moment is the number density, n(r,t) = J / ( v , r , t ) d v .

(2.48)

Hence the mass density g is Q(r,t) = mn(r,t),

(2.49)

where m is the mass of a particle. Now, we can refer to a microscopic description in terms of / ( v , r, t) and to a macroscopic description with the use of g{r,t). Kinetic phenomena are described by the Vlasov equation (e. g., Dendy 1990, Nocera, Mangeney 1999), fi,t + v • V/i + -^-[E + (v x B)] • / i i V = 0,

(2.50)

77l{

where fi(r, v) is the distribution function of the i-th species. The electric E and magnetic B fields are described by the Maxwell's equations which contain the source terms coming from the moments of all plasma species VxE

=

-B,t,

(2.51)

-VxB V V B

=

^ e * f fivdv + eE,t, J i 0,

(2.52) (2.53)

Y^eiffidv.

(2.54)

eV-E

=

In evaluations of Eqs. (2.51)-(2.53) it is important to decide what kind of electromagnetic field is needed. For strong magnetic field we can neglect magnetic field perturbations at all frequencies. In this case, electric charge density fluctuations are important and the electrostatic field is evaluated from Eq. (2.54). Both ions and electrons are often represented in this case kinetically with the relevant spatial and temporal scales equal to the electron Debye length and the inverse of the electron plasma frequency, respectively. For frequencies lower than the ion plasma frequency, electrons are treated adiabatically with the electron number density, ne = neo exp (j^f-)

-

(2-55)

Mathematical description of fluids

18

where <j> is the electrostatic potential such that E — —V<^>, and ne0 is the ambient electron density. Perturbations which satisfy this constraint are called electrostatic. For finite magnetic field, there are various regimes which can be classified in dependence on the wave frequency in comparison to the electron cyclotron frequency Qe. For frequencies much higher than the electron cyclotron frequency, both electrostatic and electromagnetic oscillations are present but they are generally weakly coupled. For a description of the electrostatic oscillations the electrostatic approximation can be used. If, on the other hand, electromagnetic waves are of interest, the Maxwell's equations have to be solved instead. For frequencies close to the electron cyclotron frequency Cle of Eq. (2.43), dynamics of electrons have to be taken into account. However, ions can be treated as an immobile fluid for appropriate time-scales. For fij0
2.7

Quasi-particle approximation

Equations (2.51)-(2.53) are difficult for analytical and numerical treatments (e. g., Klimas 1987). The usual numerical approach is to represent fa by a number of quasi-particles. Then, the Vlasov equation is solved by the method of characteristics (e. g., Tanaka 1993a). The collective behavior of the fluid can be extracted as appropriate averages over the quasi-particles which number N in the system is always lower than the actual number of particles in the system (e. g., Book 1981, Brackbill 1991). Quasi-particles positions can be found by integrating the trajectory equations: •^=Vi(t),

i = 1,2, • • - , # .

(2.56)

Here Vj is the velocity of each quasi-particle i, which can be found by integrating a force law m

i

^ = Fi({Tl},{yl},t),

J = 1,2, • • - , # , l?i,

(2.57)

Magnetohydrodynamic approximation

19

where the force Fj acting on the i-ih quasi-particle depends on the positions {r;} and velocities {v/} of all the other particles. The mass density g is found by an averaging procedure that smoothes out the discreteness, generally some form of area-weighting of finite-size particles among several cells, viz.

0(rO =< 2 m A T J -'<)>>

(2-58)

3

where the summation is done over the masses rrij of all the particles j at or near the i-th location. Similarly, the momentum density gv is found by summing the particle momenta, gv(n) =< ^2 rrijWjSivj - n) > .

(2.59)

j

When the number of quasi-particles becomes high it is useful to introduce a potential <j) such that Ft = - V 0 ( r 4 ) .

(2.60)

Quasi-particle approximation was devised for a computational treatment. However, it suffers from a drawback that the incompressibility condition is difficult to enforce self-consistently and rigorously. The quasiparticles cannot pass each other, except as a result of finite time-step errors. The numerical methods are usually stable and guarantee positivity as well as conservation. As a consequence of limited power of the computers and finite number of quasi-particles these methods may fail in accuracy and efficiency. Kinetic models were widely used to investigate physical phenomena throughout the Earth's magnetosphere (e. g., Matsumoto, Omura 1993, Winske, Omidi 1995, 1996) and low-frequency electromagnetic phenomena in kinetic three-dimensional plasma (Tanaka 1993a). The implicit algorithm was applied for the Maxwell's equations and equations of motion for the ions.

2.8

Magnetohydrodynamic approximation

A general derivation of the equations for a dynamics of plasma begins from the Boltzmann equation which describes the behaviour of plasma particles

20

Mathematical

description

of fluids

(e. g., Shizgal, Hubert 1989). By taking moments of this equation with respect to velocity, electric charge, etc., we can obtain various equations for fluid quantities such as the momentum and the electric charge density. These equations have to be combined with Maxwell's Eqs. (2.37)-(2.40). Prom Eq. (2.40), we see that

(Ml)

f~f

where / is a characteristic length-scale of the plasma, r is a characteristic time-scale. Comparing the two terms in the right hand side of Eq. (2.39), we find that eEt

El

1 El

,nM.

With the use of Eq. (2.61) this ratio can be rewritten as:

ill

c2 r 2 ' Hereafter, we drop the term —eEit in Eq. (2.39). This is justifiable if the characteristic speed 1/T, with which the considered plasma phenomenon occurs, is much lower than the speed of light c. So, Eq. (2.39) can be rewritten as j ~ - V x B. (2.63) A* It is noteworthy that Eq. (2.38) is not needed in the further analysis since this equation contains the electric charge density q which does not stand in the other equations. Moreover, with the use of Ohm's law (2.41), induction Eq. (2.40) can be written as B i t = V x ( v x B ) - V x ( i ) V x B),

(2.64)

where the magnetic diffusivity (resistivity) r} = -^. By taking the divergence of this equation we find ( V - B ) , t = 0.

(2.65)

This equation implies that the conservation of magnetic flux V • B = 0 needs only to be imposed as an initial condition. The magnetohydrodynamic equations represent coupling of the Euler equations with the Maxwell's equations under assumption that the plasma

Magnetohydrodynamic approximation

21

is treated as a single fluid, relativistic effects are neglected, most of the plasma properties are assumed isotropic, the displacement current and the separation between ions and electrons are neglected, and that we are interested in variations on a slow time scale (e. g., Priest 1982). The momentum equation contains the additional term which comes from the Lorentz force j x B , i. e. (ev)it + V • {(gv) v) = - V p + j x B .

(2.66)

The magnetic field also alters the energy equation which can be written

E,t + V-((E+p+

f - ) v - - B ( v • B ) ) - 0,

(2.67)

where the total energy E is now B2

„2

(2 68)

+

'

*-*VT 5 -

The three terms comprise the internal, kinetic, and magnetic energy, respectively. In summary, for the ideal plasma, for which rj = 0, in the Eulerian and conservative pictures the MHD equations can be written as follows: Q,t + V • (ev) = 0,

(evh + v • ({ev)v + (p + |£)i - ^ B B ) = o, B, t = V x (v x B ) , V • B = 0, E,t + V-((E+p+

| ^ ) v --B(v

B)\

= 0,

(2.69)

(2.70) (2.71) (2.72) (2.73)

where I is the unit matrix, g is the mass density, v is the velocity, B is the induction of a magnetic field which exerts the magnetic pressure J52/(2/u) and the magnetic tension - B B / / i , p is the gas pressure, and the ratio of specific heats is denoted by 7. Very often in analysis it is convenient to work with non-conservative MHD equations which can be written in the following form: Q,t + V • (ev) = 0,

(2.74)

22

Mathematical

description

of fluids

Qv,t + Q(y • V)v = - V p + - ( V x B) x B ,

(2.75)

B, t = V x (v x B ) ,

(2.76)

V - B = 0,

(2.77) (2.78)

P,t + v • Vp — —7pV • v.

2.8.1

Lagrangian

picture

In the Lagrangian picture the MHD equations can be rewritten as §

= -V-0?v),

(2.79)

dv 1 Q— = - V p + - ( V x B) x B ,

(2.80)

at

fj,

^ = (B • V)v - BV • v, dt V • B = 0,

(2.81) (2.82)

JE = _ 7 p v • v,

(2.83)

where the Lagrangian derivative is

1 = |+V-V2.8.2

Incompressible

( ^

limit

In the incompressible limit 7 -¥ 00 and V • v -4 0, such that dp/dt = ~7pV • v remains finite. Then, from Eqs. (2.79) - (2.83) we obtain

wv l 0— = - V p + - ( V x B) x B ,

(2.86)

^

(2.87)

= (B-V)v,

V • B = V • v = 0.

(2.88)

Magnetohydrodynamic approximation

2.8.3

Cold plasma

23

limit

In the cold plasma limit the pressure terms are negligibly small in comparison to magnetic terms and the former ones can be put to zero. The cold MHD equations attain the following form: Q,t + V-{gv)

=

0v, t + 0 ( v - V ) v

=

B, t V B

= =

0, -(VxB)xB, A1 V x (v x B), 0.

(2.89) (2.90) (2.91) (2.92)

Chapter 3

Linear waves

Fluid is capable of supporting a wide range of oscillations. Specifically, in plasma these oscillations may be exceedingly complex (e. g., Dendy 1990). Considerable theoretical studies have been made of three particular types of waves in plasma: magnetohydromagnetic waves, electromagnetic waves and electrostatic waves. For different types of simplifying assumptions we obtain different types of plasma waves. For example, magnetohydrodynamic waves appear only in the presence of a magnetic field, and then only for frequencies small compared with the cyclotron frequency of ions. If the electric field E is perpendicular to the direction of wave propagation k the electromagnetic waves result. The frequency of these waves is LJj_ — y/u)2 +

C2k2,

where n0e2 We = \

y eme is the electron plasma frequency, no is the ambient plasma number density, m e is the mass of the electron and e its electric charge. If, on the other hand, E and the electric current density j are parallel to the direction of propagation then longitudinal electrostatic waves occur. The low-frequency ones, with frequencies less than the ion plasma frequency, are called ion-acoustic waves. Essentially they are sound-like waves propagating in a medium whose particles have the temperature of 25

26

Linear waves

the electrons and the mass of the ions. There frequency is r2k2

2

_

^ - l

c K

s

^

i i2

+ A 2 n Jfc2 <W e'

where cs =

kBTe

is the ion-sound speed, XD

=

eTekB e2n0

is the Debye length, mi is the mass of an ion. These oscillations exist only in plasma with Te » T{. High frequency parallel oscillations are called Langmuir oscillations. They have the squared frequency a;f,=a; 2e (l + 3 A ^ 2 ) which is higher than w,22

3.1 3.1.1

Waves in homogeneous fluids MHD

waves

The presence of magnetic fields in an ionized gas affects the plasma in a number of ways; their presence exerts a force on the surrounding medium, enabling it to either maintain a state of equilibrium against external forces such as gravity, or displaces the medium, including flow. Perturbations of the field gives rise to waves which are not normally present in a hydrodynamic system. These waves are called: fast, slow and Alfven waves. Each wave has different restoring forces with which they are associated. The magnetic tension of the field line drives Alfven waves which propagate essentially along the field direction, much like waves on a string. The magnetic pressure and the gas pressure are the restoring forces for fast and slow magnetosonic waves. MHD waves are of a sophisticated nature making the sound wave look rather dull. The study of the MHD waves is a vast and fascinating subject;

Waves in homogeneous fluids

27

we will only briefly outline it here. The reader is referred to the book by Priest (1982) for more details. We consider a uniform equilibrium £o,

v 0 = 0, po,

B 0 = B0i,

(3.1)

where z stands for the unit vector in the z-direction and all quantities with the subscript 0 are constants. Assuming that perturbations are small we may linearize ideal MHD equations around this equilibrium. The linearization is based on the use of the expansion Q

=

V

=

p B

Qo + Sg,

(3.2)

<5V,

(3.3)

-

p0 + 6p,

(3.4)

=

B + <5B,

(3.5)

where the quantities with S denote perturbations. Now, we substitute these equations into the MHD Eqs. (2.69)-(2.73) and neglect quadratic and cubic terms in the perturbed quantities. Simplifying the notation by dropping S, the linearized mass continuity equation can be written as follows: gtt + goA = 0,

A = V • v.

(3.6)

«,v,* = - v ( p + ^ ) + ^ B „ ,

(3.7)

The other equations are:

B, t = - B 0 A + B 0 v , „ V • B = 0, 2

P,t = c sQ,t.

(3.8) (3.9) (3.10)

Here

is the sound speed. The terras B0Bz/p, and BoBz/p, denote a perturbed magnetic pressure and magnetic tension, respectively. Differentiating Eq. (3.7) with respect to time and substituting the other equations, we obtain v,„ = c)VA - VlVvz>z

+ Vi(v,, - Az),„

(3.11)

28

Linear waves

where we have defined the squared Alfven speed B2 y% =

—

VQo and the squared fast speed

c}=cl+Vl The ^-component of Eq. (3.11) yields t/«,« = c j A , „ . Taking the divergence of (3.11), we get

(3.12)

A,tt = c2fV2A-V^2vz
(3.13)

Differentiating this equation twice with respect to time, with the use of (3.12), we obtain (df - c}d2V2 + Vlc2sd2V2)A

= 0.

(3.14)

A Fourier mode A=

A0ej<-k-T-ut)

substituted to this equation leads to the dispersion relation A 0 (w 4 - c)k2u2 + c2sV%k22h2) = 0.

(3.15)

Here j is the imaginary unit such that

f = -1. This notation for the imaginary unit will be used throughout this book. 3.1.1.1

The Alfven wave

Equation (3.15) can be solved by Ao = 0 which implies A = 0. This equation corresponds to the Alfven wave which is incompressible. From Eq. (3.12) it follows that vz = 0. So, there is no flow along the ambient magnetic field. From the z-component of Eq. (3.8) we have Bz = 0. So, the Alfven wave is a transverse wave. Hence, the perturbed magnetic pressure B0Bz/(i — 0. The Alfven wave makes the magnetic field lines vibrate like a string, under the magnetic tension force only.

Waves in homogeneous fluids

From Eq. (3.11), for A = 0, we get that vj_ = (vx,vy,0) following wave equation: v i , « = V%v±tZZ.

29

satisfies the

(3.16)

A similar equation can be derived for the magnetic field Bj_ = (Bx,By, 0). Hence, we conclude that if the Alfven wave propagates in the k-direction its phase speed is

Hence, for the parallel propagation

On the other hand, for the perpendicular direction Bo • k — 0, the phase speed is zero and the Alfven wave cannot propagate there. So, the Alfven wave is a strongly anisotropic wave, propagating essentially along magnetic field lines. 3.1.1.2

Magnetosonic waves

The other solution of Eq. (3.15) is wi - c)k2u2 + c2sV%k2zk2 = 0.

(3.17)

This equation possesses two roots. In the case of vertical propagation, k — kz, these roots lead to

4

=

±

4c?yi

= £ \fa fo- ) •

(3 i8)

-

The solution corresponding to the phase speed c+ is called the fast wave, while c_ is associated with the slow wave. These waves are driven both by the pressure gradient and the Lorentz force. The slow wave is strongly anisotropic as it propagates essentially along magnetic field lines. The magnetosonic waves are not dispersive in homogeneous plasma but they may be so in inhomogeneous medium such as solar coronal loops (Roberts 1991a,b, Zhelyazkov et ol. 1994). It can be proven that c - < VA < c+.

Linear waves

30

In the limit of B 0 —• 0, the Alfven and slow waves disappear and the fast wave becomes the familiar sound wave. In the incompressible limit (c s -> oo) the fast wave is removed from the system and the slow and Alfven waves propagate with the Alfven speed VA- In the cold plasma limit (c s = 0) the slow wave disappears and the system propagates the fast and Alfven waves. Along the magnetic field lines these waves move with VA3.1.2

Ion-acoustic

waves

The conventional picture of ion-acoustic waves is one in which the induced electric fields, which support the oscillations, and current, are parallel to the direction of wave propagation. Then, the highly mobile electrons move rapidly to follow any potential field which is set up by disturbing the ions from their equilibrium positions. We consider a hydrogen plasma that at its equilibrium is composed of electrons and ions of equal density so that ne0 = rn0, where ne0 and n, 0 are the equilibrium electron and ion densities. The momentum equation for the ions may be written as follows: v, f + (v • V)v =

1 e Vpi + — (E + v x B) - i/v, Tumi mi

(3.19)

where v is the ion velocity, E is the electric field, B is the magnetic induction field, rrii and e are the ion mass and charge, respectively. The ion collision frequency v introduces the drag force — v\ and couples Eq. (3.19) to the corresponding equation for electrons (and neutrals, in general). Suppose there is no magnetic field (B = 0), the plasma is electroncollision dominated (v = 0) and the ions are cold, so that their pressure Pi = 0. Then a simple linear analysis, using the continuity equation, n M + V • (n
(3.20)

and assuming E is proportional to the electron pressure gradient (in the ^-direction, say) in Eq. (3.19), results in the wave equation vx,tt =

"*,**,

(3.21)

rrii

where vx is the x-component of v and Te is the electron temperature. Equation (3.21) describes linear ion-acoustic waves in an unmagnetized plasma propagating in the direction of the induced electric field, the x-direction,

Waves in homogeneous

fluids

31

with the ion-acoustic speed TekB Ci =

..

This is the simplest model of ion-acoustic waves that can be conceived. It may be argued that there are many features of the problem that have been neglected in the above model and so, for a more realistic picture we should attempt to include some of them. First, a more rigorous linear analysis, assuming that electron densities obey Maxwell-Boltzmann statistics of Eq. (2.55) and kBTe > \e\(j>, yields ee2

GSr-.«-*••••)•

"to

(3 22)

-

This is essentially the equilibrium studied by Karlicky and Jungwirth (1989). For wave-like solutions of the form e ^ * 1 - " ' ) , j 2 = - 1 , i. e. angular frequency w and wavenumber k we find the dispersion relation corresponding to (3.22) becomes

In this equation, the frequencies have been normalized to the ion-plasma frequency

Mi =

and lengths to the Debye length \D =

eTekB nine*

Now we consider a two-component, electron-collision-dominated plasma in which the electron inertia and ion temperature are neglected, so that me/mi -»• 0 and Ti/Te - • 0, respectively. If the plasma is permeated by a uniform magnetic field B = BQk,

32

Linear waves

then the ion dynamics is described by a combination of equations (3.19) (with v = 0), (3.20), (2.55) and the Maxwell's equations, leading to (Davidson 1972, Laedke, Spatschek 1982): n*,t + V • (njv) = 0, v,t + (v • V)v + V + Hit x v = 0, 2

V 0

=

(3.24)

e* - ni.

Here the velocity components have been normalized to Cj, the electric potential to Te/cs/e and the ion density nj to rij 0 . The ratio of ion cyclotron frequency of Eq. (2.43) to ion plasma frequency fl can also be considered as a normalized measure of the strength of the magnetic field,

n = rIl_

= ?±,

^/HUiTniCi

(3.25)

a

that is Bo/\/ni dictates the physics of the waves. Here VA is the Alfven speed. Small amplitude (linear), perturbations of the form e J ( k ' r _ w f ) about a static (V = 0) equilibrium of Eqs. (3.24) yield a dispersion relation (e. g., Infeld, Rowlands 1990) A-2 fc2

1

kf,

k

+ -^ O2-4=0>

(3-26)

or — iV or where fcy and k± are the components of the wave vector k parallel and perpendicular to E, respectively. In the magnetic-field-free case Q, = 0 and we recover Eq. (3.23). So that for small k, ion-acoustic waves are low-frequency, almost dispersionless waves, similar to sound waves in air. The same dispersion relation characterizes the ion-acoustic waves which propagate parallelly to the magnetic field (k± = 0 ) . As the perpendicular waves with k\\ = 0 exhibit the dispersion relation

"2 = ^ + iffci

(3 27)

"

the ion-acoustic waves are anisotropic. The presence of fi 2 in Eq. (3.27) results in a cut-off frequency. Free oscillations are possible for OJ2 > fi 2 . Below the cut-off frequency the oscillations are evanescent. We will see in the forthcoming part of this

Waves in inhomogeneous fluids

33

monograph that similar properties possess acoustic and internal gravity waves in gravity field.

3.2 3.2.1

Waves in inhomogeneous fluids Acoustic and internal stratified medium

gravity waves in

gravitationally

We consider gravitationally permeated fluid, ignoring magnetic effects. In the presence of constant gravity, momentum equation attains its form Qv,t +

• V)v = - V p + gg.

Q(V

(3.28)

In the case of gravity pointing in the negative z-direction, g = — gz, and for the static (v = 0) equilibrium this equation gives ~90o-

Po„

(3.29)

So, the inclusion of gravity introduces an additional force and a preferred direction. As a consequence of that wave motions are anisotropic and they are driven by a buoyancy force. Gravity imposes a length-scale in the fluid, which is called the density scale-height, H, Qo

H =

(3.30)

0o.. There is also a length-scale introduced by pressure variations, the pressure scale-height A such that A=

Po

^s

P0,z

19

(3.31)

Higher values of H and A correspond to weaker stratification. In particular, constant Qo{z) and po{z) profiles imply H, A —> oo. These spatial scales impose time-scales, defined as the time taken for a wave to pass the distance H and back again, viz. 2H/cs. The inverse of it is called the acoustic cut-off frequency 0J„

2H'

(3.32)

34

Linear waves

This frequency has a simple physical interpretation as the waves are propagating for their frequencies which are higher than w > ua. For lower frequencies, these waves are evanescent; they decay with z. Additionally, we can define the buoyancy (or Brunt-Vaisala.) frequency ujg such that 2 _ U) s

1

1,

9'

9

**-E>~4-li-

(3 33)

-

If u)g < 0 the equilibrium is unstable and convection sets in. This condition is known as the Schwarzschild criterion for instability. From this equation we get that the condition w^ < 0 is equivalent to 7A > H or gH from which we conclude that for a given density scale-height H the fluid is convectively unstable for sufficiently low pressure scale-height or sufficiently low sound speed or temperature. A physical meaning of ui2 can be explained as follows. Consider a fluid element which is displaced upwards. When ui2 > 0 this element is heavier than the displaced fluid, gravity pulls it back towards the original position and the element oscillates around its equilibrium position. On the other hand, when OJ2 < 0 the element is lighter than the displaced fluid and buoyancy force acts upwards, forcing the element away from equilibrium. In an isothermal atmosphere c8 does not depend on z. This implies that H — A = const and a

19 , ,2 2cs' 9

(1 - 7)3 2 c%

As 7 > 1 we have w2g < 0 and the isothermal atmosphere is convectively unstable. From Eqs. (3.29) and (3.31) we have dpo = -9Qodz = -—podz

= - ^ - .

(3.34)

Hence pQ{z) = po(0)e~ / % .

(3.35)

As for an isothermal atmosphere A = const from this equation and from Eq. (3.29) we conclude that the pressure and mass density fall off exponentially with height.

Waves in inhomogeneous fluids

35

We discuss now linear perturbations of the equilibrium of Eq. (3.29). With the use of the definition (3.36)

A= V v two-dimensional wave motions with v = (« s > 0,«,)e , ' ( *- I - w * )

(3.37)

are described by the following equations (Lamb 1932): Vz.z +

^

= (l-^i)A,

(3.38)

^)vz.

(3.39)

w - c28kl)A = (g2kl -

c ^ 2 A , 2 + g(^

These equations describe sound waves, gravity waves and instability or convection. Evaluating vz from Eq. (3.39) and substituting it into (3.38), we obtain

A,„+f(logc2),, + iJA,, + r

-2-c^

2

/

/•„, _

11„\ •

£ M f ( l o g c ^ - ( 7 - 1 ) 5 J A = 0.

(3.40)

We can eliminate the first-order derivative term by transformation u = c2syfaA.

(3.41)

Then Eq. (3.40) becomes u.22 +

u2-Q2

u„

-+ h 4 - i K UJ*

u = 0,

(3.42)

where ua is the stratified generalization of the acoustic cut-off frequency, w0, o£ = (l + 2 f f , , K .

(3.43)

Solving Eq. (3.42) for an arbitrary temperature profile is a difficult task. For example, for a linear temperature profile A (z) is expressed in terms of confluent hypergeometric functions (Evans, Roberts 1991). We consider here a more simple case of an isothermal atmosphere for which H = const.

36

Linear waves

Prom Eq. (3.43) we find that w2a — u\. This implies that Eq. (3.42) is solved by u(z) = u0ejk*z,

f = -1,

(3.44)

with

Hence we see that waves are vertically propagating if k2 > 0 and they are vertically evanescent for k2 < 0. Equation (3.45) can be rewritten as follows: w 4 - {uj2a + c2sk2)uj2 + c 2 w 2 sin 2 a k2 = 0,

(3.46)

where a is the angle between the wave vector k = (fcx, 0, kz) and the z-axis and k2 = k2x + k2. The two roots of this equation denote acoustic and internal gravity waves. Generally, these two waves are coupled. However, for a weak stratification they are virtually decoupled from one another and ^acoustic

— Csfcj

^gravity

— ^g S i n Q .

\o.*±t )

The acoustic wave is essentially isotropic. The internal gravity wave is not able to propagate along the ^-direction as for a = 0 we have uigravity — 0. As a consequence of that we claim that the internal gravity waves are anisotropic. 3.2.2

Sound waves in random

fields

Wave propagation in random media was a subject of investigations in the past. For instance, Li and Zweibel (1987) considered decay of the Alfven wave which propagated through a medium that contained time-dependent random density fluctuations. Lou and Rosner (1986) showed that the Alfven wave is damped owing to the time-dependent fluctuations. Kawahara (1976) proved that a time-dependent random field, that is associated with bottom inhomogeneities, leads to amplitude attenuation and increase (decrease) of low (high) frequency self-modulated surface gravity waves. Benilov and Pelinovsky (1989) provided an example of a time-dependent random media whose high (low) frequency fluctuations lead to wave amplification (damping). Muzychuk (1975) pointed out that space- and time-

Waves in inhomogeneous fluids

37

dependent fluctuations lead to a reduction of the mean field damping and eventually to enhancement of this field. Numerical simulations of sound waves propagation in time-dependent random media were performed by Juve, Blanc-Benon, and Wert (1999) whose approach was based on a use of the Helmholtz equation (e. g., Sobczyk 1985). The above presented investigations show that random fields effect both the wave amplitudes and frequencies. However, results of these investigations are very often contradictory and therefore they need further verification and explanation. This work is also stimulated by the recent results by Nocera, M§drek, and Murawski (2001) who discussed the effect of a spacedependent random mass density field on the acoustic waves. The main conclusion was that random density field of the Gaussian correlation function accelerates and attenuates sound waves. These results are in agreement with the findings by Shapiro and Hubral (1995) who showed that the acceleration and attenuation of sound waves propagating in one-dimensional media of space-dependent random mass density is independent of the propagation distance. In this context it is natural to discuss the influence of the medium, whose properties vary randomly, on frequencies and amplitudes of sound waves. 3.2.2.1

Waves in random mass density field

One-dimensional sound wave propagation in a gravity-free medium is described by hydrodynamic Eqs. (2.34)-(2.36). We assume now that the equilibrium mass density can be written as follows: Qe(x,t) = Q0 + Qr(x,t),

(3.48)

where go = const and gr is a random function such that its statistical ensemble average (e. g., Sobczyk 1985) is zero, i. e. < gr >= 0. In the limit of small amplitude waves we can linearize Eqs. (2.34)-(2.36) to obtain the wave equation: v,tt - clv,xx + ^-v,t Qe

where

= 0,

(3.49)

Linear waves

38

is the equilibrium sound speed which is a random function of the coordinate x. As a consequence of the presence of a random field we use the following expansion: v(x, t) = < v(x, t) > +v'(x, t),

= 0.

(3.50)

Here < v > and v' have the meaning of the coherent and random fields (e. g., Ostashev 1994), respectively. Substituting this expansion into Eq. (3.49), using a weak random field approximation (e. g., Howe 1971), we obtain the random dispersion relation (Murawski, Nocera, M§drek 2001) y+OO J— oo

/-+00 J—oo

LU

Q)"'

where Co = ^/IPO/QO is the sound speed in the deterministic medium, E(k,cj) is the Fourier transform T of the correlation function R(x2 —xi,t2 — ti) such that E(k,cj) = J7R(x,t), R{xX -X2,t2~ti)

-< Qr(X2,t2)Qr(Xi,ti)

(3.52) 2

> /'Q Q.

(3.53)

From Eq. (3.51) it follows that random sound waves are no longer dispersionless. Indeed, in the forthcoming part of the monograph we will find that as a result of interaction between random mass density field and the sound wave the latter can gain or loose energy and be slowed down or speeded up. These processes result in the presence of the right hand side in Eq. (3.51), a random correction to the classic dispersion relation for the sound waves. Time-dependent random mass density field As a special case of the dispersion relation (3.51) we consider a timedependent random mass density field, gr = gr(t). To simplify the notation we introduce the dimensionless variables K = kcoh,

fi

= ult,

(3.54)

where lt is the correlation time. Then, dispersion relation (3.51) attains the following form: n

» - ^ =n » / _ ^ 5 ^ ^ .

(3.55)

Waves in inhomogeneous fluids

39

Now, we introduce the Gaussian spectrum 2

u-Q °* E(H) = —e~ ,

(3.56)

7T

where a is the variance. For this spectrum Eq. (3.55) simplifies to the following form: r+oo o' 2 p-(fi'-«) 2 / ii_e

n2 n*

_ K2

=

^_fi2

Similarly as it was done by Nocera, M§drek, and Murawski (2001) ft can be expanded in terms ofCT2, viz. Q = K + <j2n2 + ••••

(3.58)

Substituting this expression into Eq. (3.57), replacing fi by K on the right hand side of the dispersion relation, we obtain 27Tfi2 = KypK l + f ( Z ( 0 ) - Z ( - 2 K ) )

(3.59)

where 1

f+°°

V* J-oo

e-Z2d£ £-

a

is the plasma dispersion function (Fried, Conte 1961). Using the expression for the plasma dispersion function for a real argument, Z(x + j0) = jVire'*2

~ 2D(x),

(3.61)

we obtain Z(0) = j'v^F,

Z(-2K)

= jV^e-iK*

- 2D(-2K),

(3.62)

where rx t2 D{x) = e~ / e dt (3.63) Jo is the Dawson integral (Press et al. 1992). Finally from Eq. (3.59), we get x2

27rfi2 = V^K [1 - KD(2K)) + j^K2

(l - e"4*2) .

(3.64)

Fig. 3.1 presents the dispersive curves which follow from Eq. (3.64). The solid line shows the real part of Q2- The dashed line corresponds to

Linear waves

' '

/

f *•

V

•

1 1

•

1

40

1

'

0

IT T 1

I i

2

I

I

3

4

5

K

Fig. 3.1 The real (solid line) and imaginaxy (dashed line) parts of (f2 — K)/cr2 as functions of the wavenumber K for the dispersion relation of Eq. (3.64). Timedependent random mass density field leads to frequency increase and amplification of sound waves. Numerically obtained values for the initial wave amplitude vo = 10 - 2 Co and the strength of the random field a = 0.1 are denoted by stars (real part) and diamonds (imaginary part). The error bars denote standard deviations. the imaginary part of ^2- As Sftf^ > 0 we conclude that time-dependent random field leads to the frequency increase with respect to the deterministic medium case and this increase is higher for higher values of K. The imaginary part of the frequency ^2 attains positive values which correspond to wave amplification. A positive complex part is a result of generation of coherent field at the expense of the random energy. A higher amplification occurs for higher values of K which represents shorter waves. It is noteworthy that while the dispersion relation for a space-dependent random field reveals frequency increase and wave attenuation (Nocera, M§drek, Murawski 2001), the results of these studies show that the sound waves are accelerated and amplified by a time-dependent random field. This case is very similar to the randomly perturbed swings: indeed, if p is independent of x then we can Fourier transform wave Eq. (3.49) in x; in this way, for each k, we will get exactly the equation for a parametrically perturbed pendulum, whose inertia varies randomly in time. For the Gaussian spectrum, we get always instability. And indeed this also corresponds to the experiment once we observe a child wiggling on swings.

Waves in inhomogeneous fluids

41

Fig. 3.2 Real (left panel) and imaginary (right panel) parts of Q j ^ - .

Wave noise As a second application of the dispersion relation (3.51) we discuss the case of wave noise. Using the dimensionless wavevector

and the dimensionless frequency 0 =

ulx/c0

the dispersion relation of Eq. (3.51) can be rewritten as follows: O2 - K2 = t o 2 r l

i

-E{K,~K:(l~n)dKd(l.

r

J-oo J-oo

Here lx is the correlation length. For the random mas density gr(x,t)

(3.65)

0 2 - K2

we define the wave noise as

E(K, fi) = —E(K)6(tt

-

flr(JiQ),

(3-66)

TV

where ftr is the random frequency which depends on K and 6(£l) is the Dirac's delta. As a special case we adopt dispersionless noise such that (lr(K) = crK,

(3.67)

where c r is the phase speed of the random noise. Moreover, for E(K) we limit our discussion to the case of the Gaussian spectrum of Eq. (3.56).

Linear waves

42

Fig. 3.3 Real (solid lines) and imaginary (dashed lines) parts of 0,2 -^f- versus K for cr — — 2 (left panel) and c r = 2 (right panel).

a

0 r.

~.

|

-J:

-3

'

-2

-=

I

'

• i

-.

•

,

:

-1

Fig. 3.4 Real (solid line) and imaginary (dashed line) parts of f^-^-o't for K = 2. We consider now a rightwardly propagating sound wave whose frequency can be expanded in terms of a2 as H = K + a2n2

+ ••••

(3.68)

Substituting Eqs. (3.66)-(3.68) into equation (3.65) we obtain

-n, = »*• r M r . - ^ y . - y y

CQI

(3.69)

Waves in inhomogeneous fluids

43

where we introduced the following notation: c±

(3.70)

±1.

It is noteworthy that the denominator in the integral becomes singular for c_ = 0 (c r = 1). This is the resonance case which we will defer to the forthcoming part of the monograph. Now, we consider the special case of c+ = 0 (cr = - 1 ) . Evaluation of the integral in Eq. (3.69) leads to the following expression: lx -^2 coh

= 7^K{2-i^K), 3 2 8TT /

cr = -l.

(3.71)

As the real (imaginary) part of Q2 is positive (negative) we conclude that the rightwardly propagating sound waves are speeded up and attenuated by the wave noise which moves to the left with its speed cr = — 1. For cr ^ ± 1 Eq. (3.69) can be rewritten as follows:

(Crk - c-/r)2e-(*-*>a

h^-^L c-c+(K-K)(k-<±Ky

•dK.

(3.72)

Hence K 2^372 47T

+

-KD(-K) 1 c+

K c_ , 1 -^ " —e +

+ (3.73)

Real and imaginary parts of U2lx/(coh) which follow from Eq. (3.73) are displayed in Fig. 3.2 versus the wavenumber K and the random phase speed cr. As the real part of ^Ix/icoh) > 0 everywhere accept the region close to cr — 1 we claim that the sound waves are speeded up by the wave noise there. Indeed, the Fig. 3.3 shows up that for cr = — 2 the real part of the frequency shift is positive for all values of K and the imaginary part of the frequency shift is negative for all values of K. As a consequence of that we claim that for c r = — 2 the sound waves are accelerated and attenuated for all K. Fig. 3.3 illustrates that for cr = 2 the real and imaginary parts of the frequency shifts are positive and the sound waves are accelerated and amplified by the wave noise. At the place when the phase speed of the wave noise equals the sound wave speed a resonance occurs. Fig. 3.4 shows this resonance for K — 2. Note that the resonance is of the l/c r -type; for c r = 1~ (c r = 1 + ) the

44

Linear waves

real and imaginary parts of the frequency shift are negative (positive) and the sound waves are decelerated and attenuated (accelerated and amplified) there. The wave deceleration and attenuation can be explained on the physical grounds as for cr — 1~ the sound wave interacts with slower propagating wave noise. This process is accompanied with an energy transfer from the sound wave into the wave noise, leading to the sound wave deceleration and attenuation. On the other hand, in the regime cr = 1+ the wave noise moves quicker than the sound wave and the energy is transferred into the latter one. As a consequence of that the sound wave is amplified and speeded up.

Chapter 4

Model equations for weakly nonlinear waves

As a set of fluid equations is too complex for analytical treatment it is useful to simplify this system by a model wave equation. In this chapter we present several ways of derivation of classic wave equations. Our strategy is to start from simplest conceivable model equations which are valid only for small but finite amplitude waves. Large amplitude waves are described by the fundamental set of fluid equations.

4.1

Inviscid Burgers equations for fast M H D waves

We consider the cold two-dimensional MHD equations of Eqs. (2.89)-(2.92) with the plasma velocity [vx,0,vz and the magnetic field B = [BX,Q,BZ]. The plasma background is assumed static (v = 0) and homogeneous with density go- The equilibrium can be written as 2 = 00, v = 0, B = B0z,

(4.1)

where 2 is the unit vector along the z-direction and BQ is the magnetic sequeuiiy, the m c Alfven .fiiiven apeeu, vA — H=|j, 'B field strength of the plasma. Consequently, speed, VA is constant over the entire region. 45

Model equations for weakly nonlinear waves

46

It is noteworthy that the system of equations (2.89)-(2.92) describes the fast waves only. The slow and Alfven waves were removed from the system by setting pressure p = 0 and requiring that d/dy = 0, respectively. The linear fast waves satisfy dispersion relation (3.18) with the sound speed cs = 0. This relation reads u2 =

V\k2.

In the following subsection, we consider weakly nonlinear waves which have small but finite amplitudes. 4.1.1

Nonlinear

interactions

We expand all plasma quantities around the equilibrium state given by Eq. (4.1). That is Q = vx =

Qo + tQ, Svx, vz—8vz,

(4-2) (4.3)

Bx

6BX,

(4.4)

=

BZ = B0 + 8BZ,

where S denotes perturbed quantities. After substitution of the above expansion into the MHD system (2.89)-(2.92) and dropping S for simplicity, we obtain

=

-(evz),z,

=

-Qvx,t

=

-Qvz,t ~ (Qo + Q)vzVz,z

B0vx>z

=

-(wzflx),,,

(4-8)

BZ

=

0.

(4.9)

Q,t + 0OVz,z Bo

Qovx,t

B ~ • ^ Hx,z QoVz,t

Bx,t

-

(4-5)

~ (00 + Q)VZVX,Z,

(4-6)

BXBXZ,

(4.7)

Note that all linear terms appear in the left-hand sides of these equations, while the right-hand sides contain the nonlinear terms. To gain qualitative understanding on the various nonlinear interactions in the system of equations (4.5)-(4.9) it is worth to carry out the following thought experiment. We start with the system in static equilibrium and at t = 0 we generate a perturbation in the normal flow, vx. The linear interaction between vx and Bx, Eqs. (4.6) and (4.8), gives rise to nonzero Bx and to the propagation of these two quantities away from the perturbation site,

Inviscid Burgers equations for fast MHD waves

47

giving rise to variations of vx and Bx with z. In the linear regime nothing else happens, so the ^-component of the velocity and the density can only be excited nonlinearly. The first nonlinear interactions come through the last term in Eq. (4.7), by which v2 is excited; this automatically gives rise to perturbations in the plasma density g via linear interaction with vz. We see that, after some time, all plasma quantities (except Bz, which remains unperturbed) become nonzero. Additional information on the evolution of the various physical parameters comes from the order of nonlinear terms. The one responsible for the excitation of vz, j^BxBxz, is quadratic in the linearly perturbed quantity Bx. The flow vz then interacts with the density, giving rise to selfinteraction of vz with itself. This self-interaction, described by the first two terms on the right-hand side of Eq. (4.7), is thus a fourth-order one. This process can be carried on to show that nonlinear terms in the equations for g and vz are of even order. Once the parallel velocity is different from zero, the right-hand sides of equations (4.6) and (4.8) contain terms of third order, fifth order, and so on, which means that the normal component of v and B are nonlinearly excited by odd order nonlinear terms. The absence of quadratic nonlinear terms in the equation for vx is a consequence of the lack of self-interaction of the fast wave at second order. 4.1.2

Modified

inviscid

Burgers

equation

As equations (4.5)-(4.9) are still too complex for quantitative analytical treatment, we now derive a model wave equation governing wave propagation in the limit of long wavelength. To do so, we introduce a moving coordinate frame which follows (with the speed of the linear wave, V^) propagating along the z-direction disturbances. In this frame, plasma quantities depend on the spatial coordinate z, which means that nonlinear waves are not strictly stationary in this frame. So, we perform the following coordinate stretching (e.
T

= EJ(^.-8dty

t = e2z,

(4.10)

where s measures the weakness of nonlinearity and s = 1 (s — — 1) corresponds to up-going (down-going) (along the z-direction) propagating waves.

48

Model equations for weakly nonlinear waves

In the development that follows we keep s arbitrary, although we will later take s = 1, for upwards propagation. Next, we expand the perturbed plasma quantities in powers of e 1 ' 2 , m,r)

= /o + £ 1 / 2 / i ( £ , r ) + e/ 2 (£,r) 4- e 3 / 2 / 3 (£,r) + • • •.

(4.11)

Since we will later retain the lower order terms only, this expansion means that we consider weakly nonlinear waves. Therefore, our approach consists of a first order improvement over the linear theory and can be incorrect for strongly (large amplitude) nonlinear waves. Substituting expressions (4.10) and (4.11) into equations (2.89)-(2.92) and collecting terms at e 3 / 2 and e 2 , we obtain after some algebra Qi = sy-vzi,

Bxi = -s^~vxi,

(4.12)

Bzi = 0,

(4.13)

with i — 1,2. Additionally, we get Q\ = vzX = 0, vz2 = 2\Tv*i-

( 4 - 14 )

Equation (4.12) confirms that Q and vz, on one hand, and vx and Bx, on the other, suffer the same order of nonlinearity. Moreover, Eq. (4.14) points out the different parity of nonlinear terms affecting those variables. Finally, Eq. (4.13), actually valid for i = 1,2, • • •, leads to Eq. (4.9). A compatibility condition for the ar-component of the momentum and induction equations at e 5 / 2 leads to the following modified inviscid Burgers equation with a cubic nonlinear term, 3 ^ U + iyS^Kl/r^0(4-15) Transforming this equation back into the original z, t coordinates and using vx~e^2vxl

(4.16)

we obtain ^•Z

+

^(1-4^)U:M

= 0

'

(4 17)

'

It is now convenient to introduce the following dimensionless variables

Inviscid Burgers equations for fast MHD waves

49

where L is a spatial scale. In these coordinates, Eq. (4.17) can be rewritten as follows v'XtZl+s(l-\v'^v'x
=0,

(4.19)

from which an expression for the dimensionless wave velocity, v'w, can be derived

i4=(l-f«?) - 1 -

(4-20)

As we mentioned above, we restrict the present analysis to small amplitude, nonlinear waves, such that v'x < 1. Equation (4.20) shows that for these values of v'x the dimensionless wave speed is always higher than one and grows with the wave amplitude v'x. The result v'w > 1 implies that nonlinearities speed up both positive (v'x > 0) and negative (v'x < 0) amplitude waves. 4.1.3

Inviscid

Burgers

equation

From Eq. (4.7) it follows that the parallel (vz) flow is driven by a quadratic term that is associated with the fast wave velocity component vx. Taking Vz ~ £vz2

(4.21)

v'z = \ v'l

(4.22)

we have

The immediate consequence of this relation is that v'z is of lower magnitude than v'x. In terms of v'z, Eq. (4.19) can be cast in the form of the inviscid Burgers equation, viz.

<,' + ( ' - ^ ) < , = 0 .

(4-23)

It is noteworthy that, while Eq. (4.19) contains a cubic nonlinear term, Eq. (4.23) possesses a quadratic nonlinear term. This fact, already noted after the visual inspection of equations (4.5)-(4.9), has direct consequences on the dependence of the wave speed on the amplitude, be it v'x or v'z. From

50

Model equations for weakly nonlinear waves

Eq. (4.20) or equation (4.23) it follows that the wave speed is -l

««=(*-ft*)

(4.24)

Consequently, positive and negative velocity amplitudes parallel to the equilibrium B are speeded up and slowed down, respectively, by the quadratic nonlinearity in Eq. (4.23).

4.2

The Burgers equation for acoustic waves in viscous fluid

In Sec. 4.1 we discussed inviscid Burgers equation which describes weakly nonlinear waves in an inviscid medium. In this section we see how these equations are modified by a presence of viscosity. By this purpose we consider one-dimensional analog of the Navier-Stokes equations of Eq. (2.29) for which thermal conduction is neglected, K = 0. These equations can be obtained formally from the ideal Euler equations (2.34)-(2.36) by setting Se = 0 and performing the transformation in the momentum equation, p —t p — fj,vtX, where fi > 0 is the viscous coefficient which is assumed to be constant and v is the ^-component of the velocity. Then, one dimensional viscous equations attain the following form: Q,t+ ((*>),* =

0,

(4.25)

g(vtt + vviX)

=

-p,x +/J.v,xx,

(4-26)

P,t + (pv),x

=

(l--y)pv,x.

(4.27)

For a homogeneous static equilibrium, Qo = const,

VQ = 0 ,

po = const,

(4.28)

the above fluid equations can be linearized by using the expansion Q = go + Sg,

V = SV, p-p0

+ dp,

(4.29)

where SQ, SV, and Sp are small but finite amplitude perturbations. Neglecting nonlinear terms, we obtain the following wave equation for the perturbed velocity Sv: SvtU - c2s6v,xx = —SvtXXt. Qo

(4.30)

The Burgers equation for acoustic waves in viscous fluid

51

Prom the right hand side of this equation it follows that waves are damped by the viscosity. We can evaluate this effect quantitatively by choosing a Fourier component Sv = Svej(kx-Ut),

j 2 = -l.

(4.31)

A direct substitution of this expression into Eq. (4.30) leads to the dispersion relation uj = ±kJc2s-£k2-3--^k2. V 4g§

(4.32) 2eo

Hence we see that the frequency u is complex with 9(w) < 0 which corresponds to wave damping. As a consequence of this, wave energy is transferred into heat and the wave reduces its amplitude in time. Moreover, for sufficiently long waves and weak viscosity, damped waves are oscillating. It occurs for pk < 2g0cs. Otherwise, the waves are decaying in time without oscillations. We introduce a reference frame which follows the right moving wave (Taniuti, Wei 1968) £ = e(x-

cst),

T = e2t,

(4.33)

where e is a small parameter which measures the weakness of damping. Additionally, we expand the fluid variables in terms of e (e. g., Murawski 1986b) / = /o + e / i + e 2 / 2 + ---.

(4.34)

Here / corresponds to g, v, and p and the index 0 denotes the equilibrium of Eq. (4.28). Now, e measures a weakness of nonlinearity too. Coordinate stretching (4.33) and expansion (4.34) is valid for weak viscosity and low amplitude nonlinear waves. Writing equations (4.25)-(4.27) in the coordinates £, r and collecting terms at e we obtain the following relations: 01 = — « i , Pi - Qocav\.

(4.35)

Expressions at e 2 lead to the Burgers equation "i, +

7 + 1.... A* — ^ i , < - 2 ^ i , « = 0.

(4.36)

52

Model equations for weakly nonlinear waves

In the coordinates x and t, with the use of equation v ~ ev\, the Burgers equation can be rewritten as T ~f~ 1

v,t + csvtX H

U>

— TO j . - -T—VIXX = 0.

(4-37)

Dropping the nonlinear term in this equation, we obtain the dispersion relation w =

c s jfc-j-^-jfc 2 . 2£>o

(4.38)

As this relation agrees with Eq. (4.32) in the limit k —t 0 the Burgers equation describes dynamics of long waves. A modification of the Burgers equation is the nonlinear diffusion equation, v,t - v(l - v) - v,xx = 0,

(4.39)

which is called Fischer's equation (Fischer 1936). This equation describes the propagation of a mutant gene, flame, and the Brownian motion. Solutions of this equation for a special wave speed were found by Ablowitz and Zeppetella (1979). Two dimensional generalization of the Burgers equation was derived by Bartuccelli, Pantano, and Brugarino (1983).

4.3

The Korteweg-de Vries equation for long waves in a cylinder

From Eq. (4.32) it follows that in the limit of inviscid fluid (fj, = 0) the sound waves are dispersionless since w = ±csfc. Dispersion can be introduced by implementing a characteristic spatial scale into the system. This can be done either by setting an additional physical effect such as gravity (see Eq. (3.46)) or by letting the waves propagate through a wave-guide. In the former (latter) case the dispersion can be called physical (geometrical). Another way of implementation characteristic scales through random fields was provided in Sec. 3.2.2. As an example of the medium with a characteristic spatial scale can serve incompressible fluid that is confined within an infinitely long circular cylinder, with walls of elastic rings. The fundamental set of equations can

The Korteweg-de Vries equation for long waves in a cylinder

53

be written as follows (Lamb 1980): A,t + (Av),x

=

v,t+vviX

0,

=

A,tt + ^(A-na2)

=

(4.40) p,x,

^-p,

(4.41) (4.42)

where A is the area of a crossection of the cylinder, a is the unperturbed radius, h thickness of a cylinder wall, gm mass density of a cylinder material, E Young's modulus, v fluid velocity, QQ = const fluid mass density, and p pressure. The symbols t and x denote time and distance along the cylinder axis, respectively. The cylinder introduces the characteristic spatial scale into the system. This scale is equal to the unperturbed radius a. As a consequence of this scale we expect waves to be dispersive. Indeed, equations (4.40)-(4.42) can be linearized by expansion A = na2 + 8 A, v — Sv.

(4.43) (4.44)

We obtain then the linear wave equation

SA,tt - J^5AtXX

= ^5A,xxtt.

(4.45)

Hence we see that the waves propagate with the speed

-V&

<446)

-

Due to the presence of the term 6AtXXu these waves satisfy the dispersion relation w2 =

2 2 crf ck

1+

. ^ * 2

(4.47)

As the phase speed u/k is a function of the wavenumber k a wave packet spreads in time and these waves are called dispersive. Waves in cylinders have been studied since the time of Thomas Young (1808) in connection with modeling the propagation of the arterial pressure pulse. A theory for a thin tube was presented by Moodie, Barclay,

54

Model equations for weakly nonlinear

waves

and Tait (1983). The fluid was assumed inviscid and a one-dimensional theory was extracted by averaging quantities over the tube cross-section. The other model was employed to study the propagation of pulses along initially uniform tubes and the subsequent interaction with various junctions which are characteristic of the arterial system (Moodie, Barclay, Tait 1984). Wave propagation and shock formation in nonlinear elastic and viscoelastic fluid filled tubes for a Mooney-Rivilin material was discussed by Tait and Moodie (1984). Two-dimensional analysis was performed for pulse propagation in a thin elastic tube which contains an inviscid and incompressible liquid by Barclay, Moodie, and Tait (1984). A viscoelastic shell theory for transient pressure perturbations in fluid filled tubes was constructed and tested against experiments involving water filled tube by Moodie et al. (1984). The above mentioned models suffer from the drawback that linear theories were applied. In this part of the book we present a weakly nonlinear wave theory which is devoted on the basis of the Lagrangian method. Guided by this purpose we rewrite equations (4.40)-(4.42) in dimensionless form A[t, + (A'v'),x,
=

0,

(4.48)

= =

-Ks" p' + l,

(4-49) (4.50)

where the dimensionless quantities are denoted by primes, i. e. A = na2A',

Eh p = —p',

x = lx', t = Tt', l = .H^, V 2go

I v = -v', T=xf^a.

(4.51)

(4.52)

V E

Henceforth we drop the primes in equations (4.48)-(4.50) and introduce the velocity potential rp such that v = tp,x.

(4.53)

Elimination of p and substitution (4.53) into equations (4.48)-(4.50) leads to a coupled set of equations A,x),x = 0,

(4.54)

The Korteweg-de Vries equation for long waves in a cylinder tt + \
55 (4.55)

The Lagrangian L for these equations is L =

l

-A^J

+ Att

- \A/

+ \{A - If.

(4.56)

We introduce now the moving reference frame Z = e1'\x-t),

T = e*lH,

(4.57)

where e is a small parameter which measures a weakness of dispersion. It is defined as the ratio of the diameter of the cylinder, 2a, and the wavelength A, viz. e= - .

(4.58)

The quantities A, ip, and L are expanded as A = l + eA1+e2A2 1

+ ---,

(4.59)

2

s^

= er/f! + e ip2 + ••-,

(4.60)

2

L = eLi + e L2 + •••,

(4.61)

where now e measures the weakness of nonlinearity (Taniuti, Wei 1968). In the coordinates £ and r Lagrangian (4.56) becomes L=±eArl>,i2-e1'2Ail>tt+e*/2Aii),T-he{A,e-eA,T)2

+ (A-l)2].

(4.62)

Now, we substitute expansion (4.59)-(4.61) into (4.62). Collection of terms at the same powers of e leads to the following expressions: Li = -Vi,«,

(4.63)

L2 = ^ V i / - ^iV»i,€ + VV + \Al

(464)

Ls = ^iV-1,5 2 + An/>ltT - \A1A2.

(4.65)

Prom the Euier-Lagrange equation (e. g., Infeld 1981) 6L2 SAi

dL2 dAx

d

dL2 = 0 d$dAiti

(4.66)

Model equations for weakly nonlinear

56

waves

we have Ai

(4.67)

=^itV

The Euler-Lagrange equation

Hi leads to the same result. With the use of Eq. (4.67), the Euler-Lagrange equation for L 3 takes the following form:

0=

~ W i = 2 ^ T + h*1*)*+^««-

{4 68)

-

Taking Eq. (4.53) into account we can rewrite this equation in the coordinates x and t as the Korteweg-de Vries (or shortly KdV) equation 3 1 v,t + viX + - w , x + -v<xxx

= 0.

(4.69)

From this equation after dropping the nonlinear term 3vvtX/2 we obtain the dispersion relation u = k-

\k3.

(4.70)

Indeed, this is the expression which can be obtained from Eq. (4.47) in the limit of k -> 0. As a consequence of that the KdV equation is valid for long waves only. The dispersion relation of Eq. (4.70) justifies the factors e 1 / 2 and e 3 / 2 in the coordinate stretching of Eq. (4.57). The above way of derivation of the KdV equation was presented by Murawski (1985). See also Murawski (1986a) and Murawski (1988a).

4.4

Few modifications of the K d V equation

The KdV equation was also derived in the other contexts too. For instance, Hashizume (1985) derived it for acoustic wave propagation in a straight thin elastic tube which contained an inviscid incompressible fluid. This equation was modified by Hashizume (1988) to describe wave propagation in the tube

Few modifications of the KdV equation

57

of slowly varying radius as a simple model of an artery. The fluid equations and the shell theory were applied to derive the perturbed KdV equation v,x + vv,t + v,Ut = a0v + a2v,tt-

(4-71)

In the case of viscous plasma the KdV equation is modified by the presence of second-order derivative term and it is called the KdV-Burgers equation (Murawski 1987b) v,t + vtX + fi2vvtX + a3v,xxx

+ a2v,xx = 0.

(4.72)

The exact solutions of this equation were found by Jeffrey and Xu (1989). It may occur that the nonlinear coefficient ft2 becomes zero for some values of dependent parameters. Then, higher-order nonlinear terms have to be taken into account. In the lowest-order the KdV equation contains a cubic nonlinear term v2v>x and is called the modified KdV (mKdV) equation {e.g., Murawski 1987a) v,t + u,aj + Pzv2v,x + a3vtXXX = 0.

(4-73)

In the case of the dispersive coefficient a3 = 0 higher-order dispersive terms have to balance the nonlinearity leading to the higher-order KdV equation v,t + vtX + 03v2v,x + a5vjXXXXX

= 0.

(4.74)

The combined KdV-mKdV equation, for long surface-gravity waves in a two-layer fluid was derived by Funakoshi (1985). For an inhomogeneous medium the KdV equation has variable coefficients. Such equation was derived, for instance, by Cai and Shen (1985) for surface-gravity (water) waves propagating along a channel of variable cross section. The cylindrical (n = 1) KdV, v,t + | r - vnvtT - vtrrr = 0,

r2=x2+y2,

(4.75)

and mKdV (n = 2) equations were derived by Maxon and Viecelli (1974a), Maxon (1976,1978) and Hase, Watanabe, and Tanaca (1985). For a derivation of spherical KdV equation which contains the term v/t instead of v/2t, see Maxon and Viecelli (1974b).

58

4.5

Model equations for weakly nonlinear

waves

T h e Zakharov-Kuznetsov equation for strongly m a g n e tized ion-acoustic waves

The weakly nonlinear ion-acoustic waves (in a magnetically field-free region and described in a Cartesian geometry) are governed by the KadomtsevPetviashvili (KP) equation (two-dimensional disturbances) and the KdV equation (one-dimensional disturbances) - see, for example, Infeld and Rowlands (1990). For the KP equation see Sec. 4.8. When fi = VA/C;, given by Eq. (3.25) is considered to be finite and non-zero the nonlinear dynamics of equations (3.24) may be reduced to an equation due to Laedke and Spatschek (1982) and Infeld and Rowlands (1990) u,t + yUjXXX + uu, x + - ( 1 + £l2){dtXX + Sl2)~x{utXyy

+ utXZZ) — 0, (4.76)

where u(x, y, z, t) describes the first-order of the electric potential, ion density or velocity component along the magnetic field. It can be shown that in the magnetic-free field limit (fi = 0), this equation reduces to the KP (or KdV) equation, whilst in the strong-field limit (fi ~ 1) it results in the Zakharov-Kuznetsov (ZK) equation given originally by Zakharov and Kuznetsov (1974). In canonical form the ZK equation is utt + uu,x + « , J M + utXyy + utXZZ — 0. 4.6

(4-77)

The nonlinear Schrodinger equation for modulational waves in a cylinder

In Sees. 4.2-4.2 we derived the model wave equations which are valid for long-wavelength waves. Now, we consider short-wavelength waves which are self-modulated. Such waves can be observed, for example, at a sea-shore. As a consequence of interaction with its neighbors every 10-11th wave is of a highest amplitude. This process of waves modulation is described by the nonlinear Schrodinger equation which can be derived by several methods. We choose the derivative expansion method (Kawahara 1973) and apply it to waves in the cylinder. The idea of the method is to introduce the multiple spatial and temporal scales xn=enx,

tn = ent,

n = 0,l,2,---,

(4.78)

The nonlinear Schrodinger equation for modulational waves in a cylinder

59

with XQ = x and to = *• Here e is a small parameter. As we consider the waves which propagate along the cylinder, e is defined by Eq. (4.58). The speed v and the area A are expanded in powers of e as A v

=

l + eA1+e2A2

=

2

+ ---,

(4.79)

evi+ e v2 + ••••

(4.80)

Substitution of the above scales and expansions into equations (4.54) and (4.55) leads to a sequence of expressions by equating the coefficients as powers of e. At e we obtain ^l,to+"l,zo=°>

(481)

«l,*o+^l,xo+^l,*oto*o=°-

(482)

These equations may be solved to get A1 = u(x!, x2, • • •, h, t2, • • -)eje + c.c. + ax (xi, x2, • • •, ti, t2, • • •), (4.83) i/i = -ueje

+c.c. + 0i(x1,x2,

•••,ti, t2,---),

(4.84)

2

j2 = -l,

9 = kx0-ujt0,

k u2 = Y^k~2-

(485)

Here c.c. stands for the complex conjugate to the proceeding term, u is a complex function which describes a wave envelope, ai and j3i are real functions which presence is associated with interaction between long wave and wave trains (Kawahara, Sugimoto, Kakutani 1975). We neglect this interaction by setting a\ = 0\ = 0. The solutions of the equations that are obtained by collection of terms at e 2 are A2 = 2U2u2e2i" + cc- + Kxux2,-

• • ,*i,*2,• • -)eje + c.c.+

c(xi,x2,---,ti,t2,---),

V2

=

Uil k2) 2 2 e 2j u e i

+ ex. + J(xux2,-

g(xi,x2,---

(4.86)

• • ,h,t2,

,h,h,--4)>

• • .)e?e + c.c.+

(4.87)

Model equations for weakly nonlinear

60

waves

where b, f, c, g satisfy the following relation: j{ub - kf) = uM + ^u, X l

(4.88)

and u is a solution of the characteristic equation u,tl +u,kU,Xl = 0 .

(4.89)

Prom the equations at e 3 we obtain the nonlinear Schrodinger (or NS) equation juM

+ au, ClCl + fi\u\2u = 0,

(4.90)

where: 1 a

4.6.1

0

= r ^

w3(12A;6 + 35A;4 + 39A;2 + 9)

,

"=—4**(**+3*»+3)—'

Few remarks

on the NS

<*=*!-»&•

equation

A nonlinear dispersion relation for the NS equation was obtained with the use of the average variational principle (Bhakta 1988). Stability of its solutions was discussed by Murawski and Storer (1989). Modified NS equation with the exponential nonlinear term was derived and studied for stability of its solutions by Murawski (1991a) and Murawski (1991b). The NS equation with additional term that contains u was derived by Ono (1991) for self-modulation of surface-gravity waves propagating above spatially inhomogeneous bottom. An extension of the NS equation on non-ideal fluid leads to the GinzburgLandau equation (Stewartson, Stuart 1971, Yang 1990), ju,t + (1 - jco)n,xx + (1 + j-)\u\2u

- j^-u

- 0.

(4.91)

This equation describes the evolution of the wave envelope u in various problems such as Raleigh-Benard convection (Newell, Whitehead 1969), Taylor-Couette flow (Kogelman, DiPrima 1970), drift dissipative waves in plasma (Nozaki, Bekki 1983), and turbulent motion in chemical reactions (Kuramoto, Yamada 1976). In the limit of zero-dissipation CQ = 0 and the NS equation is obtained. The equation with CQ ^ 0 possesses solitary wave, hole, and shock-type solutions (Nozaki, Bekki 1984). The numerical results, performed with the use of the Galerkin scheme (e. g., Wendt 1992),

Few other model wave equations

61

reveal that in the limit CQ = 0, the solutions of Eq. (4.91) approach those of the NS equation.

4.7

Few other model wave equations

In this part of the book we mention few other model equations. We start with the Boussinesq equation (Boussinesq 1872), Ujt ± U,xx ~ 6(u2),xx

- U,xxxx = 0.

(4.92)

This equation describes small amplitude bi-directional waves; these waves can propagate in two opposite directions. Solutions of this equation were found by Nakamura (1979), Murawski (1988b), and Tajiri and Murakami (1991). Breather solutions, which consist a combined state of two solutions, were obtained by Tajiri and Murakami (1989). The Regularized Long Wave (RLW), known also as the Benjamin-BonaMahoney, equation, u,t + uujX - u,xxt = 0,

(4.93)

was first proposed by Peregrine (1966) as an alternative model to the KdV equation. This equation has certain advantages over the KdV equation (Benjamin, Bona, Mahoney 1972). It possesses solitary wave solutions similar to those of the KdV equation (Eilbeck, McGuire 1977). Their stability was examined by Murawski (1989). The Benjamin-Ono (BO) equation (Benjamin 1967), u,t + u,x + uutX + HujXX = 0

(4.94)

was derived for a compressible deep fluid by Miesen, Kamp, and Sluijter (1990). Here H is the Hilbert transform, , 1 f°° r r , Hu(x,t) = -P *" J-oo

u(s,t)ds K

' , S-X

where P stands for the principal value. The BO equation describes long internal gravity waves in an incompressible, stratified fluid with a density that varies only in a layer whose thickness is much smaller than the total depth.

62

Model equations for weakly nonlinear

waves

The Zakharov equations, iE,t+E,xx

=

nE,

(4.95)

n,a-ntXX

=

\E\\xx,

(4.96)

were originally derived by Zakharov (1972). The first equation describes the evolution of the envelope of Langmuir waves given by E(x, t) with the nonlinearity introduced by the term containing the density fluctuation n(x, t) which evolution is governed by the second equation with the ponderomotive force exerted by the Langmuir wave. Stability of periodic waves and soliton solutions of these equations were discussed by (Murawski, Ziemkiewicz, Infeld 1991). The interaction of two waves ui and u2 which propagate with speeds c\ and c2 can be described by the following equations (Hasegawa 1974): wi,t + ciui ia;

=

-uiu2,

(4.97)

U2,t + C2U2,x

=

uiu2.

(4.98)

Coupled nonlinear wave equations which describe an interaction between short and long capillary-gravity waves on a liquid layer of uniform depth were derived by Kawahara, Sugimoto, and Kakutani (1975).

4.8

Remarks on multi-dimensional wave equations

Among multi-dimensional wave equations the most known equation is the Kadomtsev-Petviashvili (KP) equation, (u,t + uutX + u,xxx),x

± um

= 0,

(4.99)

which is a two-dimensional analog of the KdV equation (Kadomtsev, Petviashvili 1970, Taha, Ablowitz 1984). This equation models waves propagating primarily in one direction with only weak variation in the orthogonal direction. The Kadomtsev-Petviashvili-Burgers equation, (utt + uutX + utXX + uiXXX)>x ± utyy = 0,

(4.100)

for shallow water waves was derived by Bartucelli, Muto, Carbonaro (1984) and Bartucelli, Carbonaro, Muto (1985).

Remarks on multi-dimensional wave equations

63

Solitons and rational solutions of the KdV, modified KdV, Boussinesq, and KP equations were obtained with the use of the direct method by Ablowitz and Satsuma (1978). Special periodic (in the {/-direction) solutions of the KP equation with positive dispersion (minus sign in Eq. (4.99)) were found by Tajiri and Murakami (1989, 1990). It is noteworthy that the KP with positive dispersion do not possess stable soliton solutions. A packet of such waves either collapse or spread out. Model wave equations for vortices were derived by Petviashvili and Pokhotelov (1992). These vortices can be maintained by winds or zonal flows. They are also supported by heat flux from the condensation of supersaturated vapor in the Earth's atmosphere near the surface of oceans or by the buoyancy force which can occur during sandstorms in deserts. At high altitudes these vortices can be also enhanced by heat released in exothermic reactions. They are analogous to plasma drift waves which are described by Hasegawa-Mima (1978) equation. As in a temporal development of these vortices the Coriolis force is important, they are called Rossby waves. A good example of such waves is the Great Red Spot of Jupiter (Petviashvili, Pokhotelov 1992) which is an anticyclon soliton, disappearing and reappearing 17 times since it was discovered in 1664 by Robert Hooke.

Chapter 5

Analytical methods for solving the classical model wave equations

5.1

Analytical solution of the inviscid Burgers equation

In Eq. (2.16) we can choose f{u) = u2/2 to obtain the inviscid Burgers equation which is written here in the canonical form u,t + uutX = 0.

(5.1)

Somehow modified version of this equation was derived in Sec. 4.1.3 for flow along magnetic field lines. A solution of this equation, with the initial condition u(x,t = 0) = uo(x), is given by the implicit equation u = uo(£)>

£ = x -ut.

(5.2)

Hence by evaluating temporal and spatial derivatives u

,t = u
(5-3)

utX - tto,£(£,* + £,uu,x),

(5-4)

and the use of Eq. (5.1), we obtain (1 + tu0£)(utt

+ uu,x) = 0.

(5.5)

So, for x, t, and u 0 such that 1 + tuo^^O,

(5.6)

the solution of the inviscid Burgers equation is indeed given by u — UQ(X — ut). 65

66

Analytical methods for solving the classical model wave equations

As a special case we consider the following initial condition: uo(^) = cos (irx)

(5.7)

u — cos [ix{x — ut)].

(5.8)

from which we have

Prom expression (5.6) it follows that t #

.\ v (5-9) 7rsin(7ra;) So, this equation is satisfied for all* < 1/ir = tt,. In the course of time the initial perturbation becomes steeper until it breaks at t = *&. As a result of the wave breaking, for t > tf, there are multivalued solutions but they are still described by the inviscid Burgers equation. This equation consists a convective part of the most model equations which are presented in this book. 5.2

The direct method for the Burgers equation

In order to solve the Burgers equation, utt + /3uutX + autXX = 0,

(5.10)

we apply the direct method (Hirota 1980), the idea of which is to introduce the functions f(x,t) and g(x,t) such that „(,,«,

S

£ | | ,

(5

.u)

and then to derive equations for f(x, t) and g(x, t) that are simple to solve. The following formulae are used: ' ^

/A.

g\ J / ,xx

-

D 9f

' 2

(5.12)

f '

D2xgf J

2

gD2xff J/

Jn

'

(5.13)

where the bilinear operator D™ is defined as follows:

D b

> =^x{^~^ya{x)bix,)'neN-

(5,i4)

The direct method for the Burgers

equation

67

It can be shown that D>1

= a
(5.15)

D£ab

= {-l)nDlba.

(5.17)

Substituting expression (5.11) into Eq. (5.10) and rearranging terms, we get Dtgf + aDlgf j2

JDxgf-aDlff p =0.

+9

(5.18)

Hence we have the coupled equations, (Dt + aD2x)gf 0Dxgf

= 0,

(5.19) 2

= aD xff.

(5.20)

Now we expand / and g in power series with a parameter e, oo

oo

/ =l +$>n/n,

" < ? „ ,

n=l

(5.21)

n=0

and substitute these expressions into Eqs. (5.19) and (5.20). Collection of terms at the same power of e leads to the following equations: (Dt + aD2x)(goh + gil) = 0, 2

(5.22)

(Dt + aD x)(g0f2 + gxfi +
(5.23)

pDx(g0fi

+ ^1),

(5.24)

pDx{g0f2 + 9lh + g2l) = aD x(lf2 + hh + M ) .

(5.25)

+ 9ll) = aDl(lh 2

We choose the following pair of stationary solutions of (5.22) and (5.24): /i(0=e c ,

9i(0=bet, ^ = x-ct.

(5.26)

Then, with the use of property (5.16) we deduce that all higher-order terms can be taken to be zero, fn = gn — 0, n > 2, and b and c are given as b = 2^+gQ,

c = a + Pg0.

(5.27)

Setting e = 1, the solution is then given by

.(0 = 6i£.

(5-28)

68

Analytical

methods for solving the classical model wave

equations

This solution corresponds to the shock such that u(£) -> go when £ -» — oo and u(£) ->• 6 for £ -»• oo (Fig. 5.1).

1.0

0.5

o

0.0

-0.5

-1.0 -20

0 {

-10

10

20

Fig. 5.1 The shock solution of Eq. (5.28) to the Burgers equation (5.10) for a = —1, /3 = 1, and 30 = 1. Stationary wave solutions of the Burgers equation can be also obtained with the use of classical integration (Karpman 1975, Bhatnagar 1979). The Burgers equation, u,t + uutX + auiXX = 0, can be transformed to the heat equation, w,t =

-awjXX,

via Cole-Hopf transformation (Cole 1951, Hopf 1950), u = 2a(log w),x. Tay's and Parker's solutions for this equation, which use a sum of shocks, is presented by Whitham (1984). Prolongation and moving poles methods were described by Ruijgrok (1983).

Backlund transformation for the Korteweg-de Vries equation

5.3

69

Backlund transformation for the Korteweg-de Vries equation

We present main ideas of Backlund transformation here. For details, the reader is referred to the paper by Wahlquist and Estabrook (1973) and the book by Drazin and Johnson (1989). We consider two distinct solutions u and w of the KdV equation which is now written in the canonical form u,t + I2uu,x + u,xxx = 0.

(5.29)

It is required that these solutions satisfy the PfafRan system wiX

=

P{w,u,utt,utX,utXX),

wtt

=

Q(w,u,utt,utX,u,xx).

For known u(x,t) and functions P and Q we have two uncoupled first-order equations for w(x,t), which are easy to solve. The Pfaffian system allows to progress from one solution to another one. Unfortunately, finding P and Q is not straightforward, Wahlquist and Estabrook (1973) found that P Q

w2-a2,

= =

2

2

(5.30) 2

Aa {a -w ),

(5.31)

the integrability condition, io, xt = wttx, leads to the equation which is satisfied by w(x,t) w,t ~ 6u>,z2 + wtXXX = 0.

(5.32)

This equation transforms into the KdV equation of Eq. (5.29) for u = —w<x.

(5.33)

Eqs. (5.30) and (5.31) can be used for construction of an unknown solution as w>x

= w2 - a2 — -u,

wtt — 4a 2 (a 2 -w2)

= 4a2u.

(5.34)

These equations can be solved to get w = —atanh[a{x — 4a 2 i)].

(5.35)

70

5.3.1

Analytical methods for solving the classical model wave equations

Solitons

With the use of transformation (5.33) we find that expression (5.35) corresponds to the soliton solution (Fig. 5.2) u{x,t) =

cosh2 [a(x — 4a2t)]'

(5.36)

The above described process can be continued by setting the known soliton solution into the Pfaffian system to obtain 2-soliton solution and further more to find n-soliton solutions (Wahlquist and Estabrook 1973). From Eq. (5.36) it follows that the soliton of its amplitude a2 propagates with the speed 4a 2 . So, higher solitons propagate quicker.

Fig. 5.2 The soliton solution of Eq. (5.36) to the KdV equation (5.29) as a function of f = x — 4a t for o = 1.

A solitary (localized) wave is called a soliton when it emerges from the interaction with the other solitary wave unchanged in shape and speed, without producing any other alteration in the vicinity of the interacting waves. Not all solitary waves are solitons. For example, a flame wave traveling down a candle eats itself. Such flame can be recognized as a solitary wave. However, two flames at either end of a candle annihilate each other when they meet together. A wave of this type was first observed in 1834 by John Scott Russell who saw a boat that stopped suddenly when he was riding his horse along

Inverse scattering

method

71

the Union Canal in Edingburgh. He followed the solitary wave that was generated at the boat's prow, until after a chase of one or two miles he lost it in the windings of the canal. Later on, Russell performed laboratory experiments in which he excited solitary waves by dropping a weight at one end of a water channel. His experiments lead him to conclude some peculiar properties of the 'great waves of translation' which speed c = \/g{h + a), where h is the unperturbed water depth, a is the wave amplitude and g is the acceleration of gravity. From this formula it follows that higher waves travel faster. As a consequence of that a taller wave can catch up a lower wave, interacts with it and then passes through, overtaking the lower one and continuing on its way intact and undistorted.

5.4

Inverse scattering method

5.4.1

Lax

criterion

We present here the inverse scattering method which is based on Lax criterion (Lax 1968). The idea of this method is to find a skew symmetric operator B such that u,t = [L,B],

(5.37)

where: L = --^+u(x,t)

(5.38)

is the Sturm-Liouville operator with u(x, t) playing a role of the potential which depends parametrically on time t. In Eq. (5.37) the operator B has to be chosen such that the commutator [L, B] is the operator of multiplication by a number. Then, this equation is equivalent to a partial differential equation. As a special case we consider the following operator: B = D5 + h{x)D + b2(x)D3 + Dbxix) + D3b2{x),

(5.39)

where we introduced the notation

^£'

n= 1 2

> .--

< 5 - 4 °)

72

Analytical methods for solving the classical model wave

equations

While evaluating the commutator [L, B] we equate to zero the coefficients at various powers of D. The coefficients at D5 and D7 are equal to zero identically. The other coeffcients lead to the following equations: 4D 2 6i + 5D% + 5D4u + 6b2D2u + 6(Db2)Du = 0, 3

3

ADbx + 9D b2 + WD u + 6b2Du = 0, 2

2

(5.41) (5.42)

4£> 62 + 5D u = 0,

(5.43)

4Db2 + 5Du = 0.

(5.44)

Hence we obtain the expressions for &i and b2 as bi = - ^ (5D2u + 12u2 - 24cu), 16 b2 = -^u + c,

(5.45) (5.46)

where c is the integration constant. Then, Eq. (5.37) leads to the modified fith-order Korteweg-de Vries equation (Murawski 1987c) u,t+luD3u-l(Du)D2u-\cD3u~u2Du+3cuDu+^D5u 8 2 2 8 5.4.2

Inverse

ID

= 0. (5.47)

method

The literature dealing with the inverse scattering method is extensive. The reader is referred to the papers by Gel'fand and Levitan (1955), Kay and Moses (1956), Gardner et al. (1967), Miura et al. (1968), Chadan and Sabatier (1977), Leibovich and Sibass (1977), Wadati et al. (1979), Novikov (1980), Fokas and Ablowitz (1983). The eigenfunctions of the Sturm-Liouville operator L can be written asymptotically as 1>(x,t) ip{x,t)

= =

a(k,t)e-ikx ikx

e~ ,

+ b(k,t)eikx,

x -4oo,

x -> - o o ,

(5.48) (5.49)

where b and a are called respectively the reflected and transmitted coefficients. As a consequence of conservation of energy these coefficients satisfy the following equation: \a\2 - \b\2 = 1.

(5.50)

73

Inverse scattering method

The quantities e~lkx and elkx correspond to the left-propagating and rightgoing waves, respectively. For a discrete spectrum the wave function is given by cj){x,t)

bn(an,t)ea"x,

=

4>(x,t) =

e~anX,

z->-oo,

x ->oo.

(5.51) (5.52)

Novikov (1980) showed that the function g{x, t) = ip,t + Brl>

(5.53)

is an eigenfunction of the operator L. In the limit x ->• — oo we obtain g(x,t) = ik(2ck2 -kA-

l)e~ikx.

(5.54)

So, the time evolution of the function ip is V>,t = ik{2ck2 - fc4 - l)e~ikx

- Bxj).

(5.55)

Hence in the limit of x -+ — oo, we find the evolution of the scattering data a{k,t) b{k,t) bn(an,t)

= =

a{k,0)e'ikt, 6(fc,0)e i * (4cfc2 - 2 * 4 - 1 )',

(5.56) (5.57)

=

bn(an,0)e2a^<+^.

(5.58)

The quantities such as a(k, 0) can be determined from the initial data for Eq. (5.47). We use now the Gel'fand-Levitan-Marchenko linear integral equation for the case of b(k, t) = 0 and with the kernel defined by the following formula (Novikov 1980):

1

d(ia„)

The solution of Eq. (5.47) is given by the formula (Novikov 1980) u(x,t) = -2D2logA,

(5.60)

where: A = i+beM~2anx)

(561)

74

Analytical methods for solving the classical model wave equations

Then, the solution of the modified Korteweg-de Vries equation is the solitary wave which is given by the following expression: "

(X l) =

'

-2a2 cos/i{a„[(2< + l)t - 2x] - logbn{0)} + 1'

(5 62)

"

This solitary wave propagates with its speed 1 _L O n , 4

n

V =

n

(5.63)

and amplitude

Uo = 5.5

4a2t&n(0)

-[rTMoF

,KRA,

(5 64)

-

Stationary wave solutions of the nonlinear Schrodinger equation

In this part of the book we discuss stationary solutions of the nonlinear Schrodinger equation. It is convenient to rewrite Eq. (4.90) in the coordinates of the moving frame which follows a wave with its speed c £ = C - cf,

T = T.

(5.65)

+ /3|u|2u = 0.

(5.66)

Then j(utf - cu^) + au^

We look now for the stationary solutions of the following form:

«(f, r) = MO exp [j ( ^ + 6f)] ,

(5.67)

where b is an arbitrary constant. This solution represents a slowly varying envelope uo(£) with a phase shift represented by the exponential term. Substituting Eq. (5.67) into (5.66), we get auo(( + (fiul + a)u0 = 0,

(5.68)

c2 a = — - b. 4a

(5.69)

where:

Stationary

wave solutions of the nonlinear Schrodinger

equation

75

Multiplying Eq. (5.68) by f uo,£ and then integrating, we arrive at = -^-u40--u2Q + l = Y(u0), (5.70) la a where I is an integration constant. The qualitative nature of the solution to Eq. (5.70) may be determined from an inspection of the function Y(UQ) which should be bounded for bounded u0 and must possess double roots (Zhelyazkov et al. 1994). These double roots occur for w0 satisfying the following equations: uo/

Y,uo=Y = 0.

(5.71)

The one and two double roots usually correspond to a soliton or a shock wave (e. g., Murawski 1987a). Other values of I for which Y{UQ) is bounded and Y{UQ) > 0 are associated with periodic waves. The double roots occur IOr lrnin,max > w U e r e

lmin,max = min, max (0, - •— J .

(5.72)

Based on the above considerations we can distinguish three families of solutions, depending on the values of a/?, c, and I. (i) The soliton family. This occurs for aft > 0 and a/3 < 0. The case of I = Imax corresponds to the bright soliton solution which is given by (e. g., Hasegawa 1989) UO(C,T) = a s e c h | \/^a

(C - br - s) ) exp I i—(C - CT) 1 ,

(5-73)

where a and b are arbitrary constants, c < 6/2, and s is a phase shift. This soliton is presented in Fig. 5.3 by the curve B. For lmin < I < Imax the solutions are periodic waves. One such wave is 2

2

(a2-a?)sn2(±a2|-£|m).

(5.74)

Here, a\ and 02 are the module of roots of the equation Y(uo) = 0 such that ai < 02; sn(y \m) is the Jacobian elliptic sine function of arguments y and m = (a\ — aD/a^- This wave is represented by the curve C in Fig. 5.3. The case of I > lmax corresponds also to periodic waves, a representative

76

Analytical methods /or solving the classical model wave equations

Fig. 5.3 The solutions of the nonlinear Schrodinger equation for a > 0, fl > 0, and a < 0. Curves A and C denote periodic waves, and curve B is associated with a bright soliton.

of which (see the curve A in Fig. 5.3) is given by un — a.

sn- | ±a2\rK

^

2

' £|m

(5.75)

with m — a\l{a\ + a%). In this case Y = - — (UQ — a?) (UQ + a | ) . (ii) The periodic wave family. This family exists under the condition of aB > 0 and aB > 0. The family contains only periodic waves, which are described by formula (5.75). This wave is shown in Fig. 5.4. (iii) The shock wave family. These solutions are restricted to the case of a/? < 0 and a/3 < 0. For / = lmax there is a shock wave (black soliton) solution. See Fig. 5.5, curves A expressed by

u0

i^vW,

(5.76)

For Imin < I < lmax there are periodic waves with (see curve B in Fig. 5.5) u 0 = ± ai sn I ± W — a 2 £ | m J ,

(5.77)

Stationary wave solutions of the nonlinear Schrodinger equation

77

/"\

V

\

/

Fig. 5.4 The periodic-wave solutions of the nonlinear Schrodinger equation for a > 0, /3 > 0, and a > 0.

\

/

/

Fig. 5.5 The solutions of the nonlinear Schrodinger equation for a > 0, /3 < 0, and a > 0.

78

Analytical methods for solving the classical model wave

equations

where a\ and a-i are the module of roots of the equation Y(uo) = 0 such that a\ < 02 and m = a\/a\. The nonlinear Schrodinger equation possesses also the dark soliton solution (Hasegawa, Tappert 1973, Hasegawa 1989) which can be expressed in the following form: U(C,T)

= J—jas

[1 - a 2 sech 2 (s/a^aaZ)] exp[i(C,r)],

(5.78)

with 0 « , T ) = Va* (1 - a 2 ) C + t a n - 1 f

^

g

tanh(v/o7ao0 J

+ a a s (3 - a 2 ) r,

|a a | < 1.

(5.79)

The special case of aa = 1 corresponds to the black soliton solution of Eq. (5.76). Solutions, representable in terms of Wronskian determinants, of the NS equation and some other nonlinear evolution equations were shown by Freeman (1984).

Chapter 6

Numerical methods for a scalar hyperbolic equation

6.1

Finite-difference approximations

This section contains a very brief overview of the fundamental ideas that arise in finite-difference discretization of simple partial differential equations. In the category of finite-difference schemes, there are many kinds of methods existing, with each having its own advantages and disadvantages. According to numerical accuracy, there are first-order, second-order and higher-order schemes. First-order schemes are known to have too much numerical diffusion. Higher-order schemes are generally more dispersive and require more CPU time than second-order schemes. However, all schemes are subject to many constraints such as causality, positivity, conservation, and time reversibility. Therefore these constraints should be built into numerical algorithms. With a finite-difference method, we discretize the variable u" = u(xi,tn),

Xi = iAx,

tn = nAt,

i = 1,2, •••, n = 0,1,- • •,

(6.1)

and its derivatives which appear in partial differential equations. By that way we obtain a system of algebraic equations for the variables evaluated at the grid points. There are many different methods to evaluate the derivatives, leading to different finite-difference methods. Few examples are provided by the text below. Finite-difference approximation of derivatives is based on a Taylor's series expansion. For example, if Ui corresponds to point i, then u;+i at point i + 1 can be expressed in terms of a Taylor's series expanded about 79

80

Numerical methods for a scalar hyperbolic

equation

point i, as follows: (Ax) 2 =Ui + u,XiAx + u,XXi— 1

u i+i

.

(6.2)

This equation contains an infinite number of terms which are convergent for sufficiently small grid size Ax. For numerical computations, it is impractical to carry this infinite number of terms. Therefore, Eq. (6.2) is truncated. For example, if terms of magnitude (Ax) 2 and higher-order are neglected, Eq. (6.2) reduces to Ui+i ~ Ui + u,XiAx.

(6.3)

We say that this equation is of first-order accuracy, as terms of order (Ax) 2 and higher are neglected. These neglected terms represent the truncation error te, i. e. (Ax) n t^E^i-^rn! oo

(6-4)

n=2

Then, from Eq. (6.2) we obtain Ui+l ~ Ui

"'•<

=

~Ax—

te

~Ax-

Ui+i =

-Ui

~Ax—

+

°(A:C)'

(6

-5)

where the symbol O(Ax) is a formal mathematical notation which represents terms of order Ax. As the derivative is computed with the use of forward value of u*+i we call this scheme as a first-order forward difference. Similarly, we can show that a first-order backward difference scheme is Ui Ui 1

~ ~ Ax

+0(Ax).

(6.6)

This scheme is obtained from the Taylor's expansion Ui-i — Ui + utXi(-Ax)

H

.

(6.7)

Subtracting this equation from Eq. (6.2) we arrive at a second-order central difference scheme ^

= Mi+

2AxMi"1+0((Aa:)2)-

(6 8)

'

Other schemes can also be built, for instance u, x{ =

g ^

+ 0{(AxY).

(6.9)

Simple finite-difference schemes

81

This and other schemes can be obtained with the use of the method of undetermined coefficients. Suppose we have utXi = aui+i + bui + cui-i + dui-2,

(6.10)

where a, b, c, and d are coefficients which need to be determined. Expanding in Taylor series Uj+i, Uj_i, u;_2 around ttj and collecting terms at m, utXi, u ,xi_i, and utXi+1 lead to four algebraic equations for a, b, c, and d. It is noteworthy that there are other methods to derive finite-difference approximations. For instance, the function u(x) can be approximated by polynomial w(x). Then, wtX is used as an approximation to u )X . The polynomial can be determined by interpolating u at an appropriate set points. To derive the scheme of Eq. (6.9), w(x) is the cubic polynomial that interpolates u at x;+i, x,, Xj_i, and x-;_2. A second-order derivative can be evaluated as -(u ) U,xx{ - \U,x),xi

u

' (ui+1

LV

~ui\

Ax J

flli-Ui-i

V

V

; j A^ =

AX

- "•*<+!""•*< — /±x —

(AxP

•

(6

-H)

The above formula for evaluation derivatives can be used for construction finite-difference schemes for advection Eq. (2.12). This will be done in the proceeding section. 6.2

Simple finite-difference schemes

The history of the development of numerical schemes for hyperbolic equations, which possess shock solutions, is long and rich. The idea that shockcapturing can be accomplished through an appropriate dissipation term was used in early numerical schemes. We explain it in some details considering the time-dependent Cauchy (initial-value) problem for advection Eq. (2.12). We have several choices for representing the time derivative term. For a forward Euler differencing scheme we have «,*? =

a

% At

+ O(At).

(6.12)

Here i = 1,2, • • • corresponds to the cell center and u is a step-wise function ofx (Fig. 6.1).

82

Numerical methods for a scalar hyperbolic

i-1

equation

i+1

i->2

i-i->2

Fig. 6.1 Spatial discretization of the function u(x). The cell center is denoted by i while its interfaces correspond to i ± 1/2. For the space derivative, we can use a second-order representation (centered Euler differencing scheme) of Eq. (6.8) still using only quantities known at old time-step n H+l

\Xi

H-i

2Az

+ 0((Ax)2).

(6.13)

Here Ax = xi+i/2 — Xi-1/2 (Fig- 6.1). The resulting (finite-difference) discrete approximation to advection Eq. (2.12) can be written as: ,,"+1 _ „ n Ui

Ui A

At

„"

_ ,.«

= -cU<+'

V l

.

2Ax

(6.14) v '

With a use of notation for the fluxes this equation can be rewritten as: A z « + 1 - < ) = -At(f?+1/2

- /f_ 1 / 2 ).

(6.15)

The numerical flux / is an approximation to the integral over time at the cell interface a; i+1 / 2 , i- e. /r+i/2 = A 7 / " + 1 c u ^ + i / 2 ^ ) d t - \ « + i + < ) •

(°6)

It will be proven mathematically in Sec. 6.5 that the above scheme, though accurate, is unconditionally unstable and therefore it is useless in practice. An unstable scheme would amplify little errors in the solution until the algorithm breaks down.

Simple

finite-difference

83

schemes

Second-order accurate scheme, both in space and time, can be obtained with the use of centered approximations to the derivatives, viz. ,.n+l „,n-l Ui - U;

.^! 0 2M - ^u, k- fu^ '

< 6 ' 17 >

i+1

which leads to the leap-frog method < + i 6.2.1

Lax-Wendroff

= u r

i - | i «

+ 1

- u t J .

and Lax-Fredrichs

(6.18)

schemes

A stable scheme can also be obtained with the use of the Taylor expansion u{x,t + At) = u(x,t) + At utt(x,t)

(At)2 + ^-^-uttt(x,*)

+ ••••

(6.19)

Moreover, we have uttt = -cutXt = -cuttx

= -c(utt),x = c2utXX.

(6.20)

Substituting this expression into Eq. (6.19), we obtain u(x,t + At) = u(x,t) -cAt

u,x(x,t) +

(At)2

c2utXX(x,t)

H

.

(6.21)

Using centered-difference approximations for the spatial derivatives (see Eqs. (6.8) and (6.11)) we get the Lax-Wendroff scheme (Lax, Wendroff 1964) <+1

=

< ~ 2^(<+> ~ <" l}

+

°^${U^

~

2 <

+ UU)

-

(6 22)

-

This equation is the forward-time centered-space discretization of the equation u,t + cu,x = —z—u,xx.

(6.23)

The term c2AtutXx/^ represents numerical diffusion which originates from the discretization. Another numerically stable scheme can also be obtained by replacement un in Eq. (6.15) by its average <-i+<+i,

(6.24)

84

Numerical methods for a scalar hyperbolic

equation

This turns into the Lax-Fredrichs scheme (e. g., Press et al. 1992) Ax ( < + 1 -

<

"

1

^

< + 1

) = -At(/f+1/2 - /r_1/2),

(6.25)

which can be rewritten in the form of Eq. (6.14) with a reminder term, i. e. _ At

< + 1

u?

^

^

.

^ 2Ax

+

i /< + 1 -2< + <_12 \ At

(6 2g)

Equation (6.26) is exactly the forward time centered space representation of the equation (Ax)2 u,t + cutX = ~^fu,xx

(6.27)

in which the right-hand side represents diffusion. However, this method is first order accurate and consequently it is seldom used in practise. This method is stable provided the time step At satisfies |cAt| < Ax.

(6.28)

This stability restriction allows us to use a time step At = 0 ( A x ) . From Eqs. (6.22) and (6.27) it follows that we have, in effect, added a diffusion term to the generalized density equation. The Lax-Fredrichs and Lax-Wendroff schemes are thus said to introduce numerical dissipation. These schemes belong to a family of old shock-capturing schemes. From Eq. (6.27), we see that the essence of old shock-capturing schemes is to smear a steep profile over a small number of grid points. The smearing is done by introducing into the system sufficient amount of a linear dissipation which is implemented globally, both in high-gradient regions and low-gradient regions. 6.2.2

The Beam-Warming

method

A second-order method can be derived by following the derivation of the Lax-Wendroff scheme in which one-sided approximations to the spatial derivatives are used instead. This results in the Beam-Warming method: <+1

rAt + ?~ 2 A * " ( 3 < " 4 u " - 1 + UU) cAt ^-(„?-2«?_1+u?_2), c>0.

=U

(6.29)

Simple

finite-difference

schemes

85

This method is stable for 0 < ^ < 2 . 6.2.3

The MacCormack

(6.30)

scheme

The MacCormack scheme for Eq. (2.16) combines forward and backward Euler schemes in separate predictor and corrector steps. There are four versions of this scheme (Hirsch 1990). One of them is as follows: a) predictor step A t , .„

,„x

^

^

«? = «? - A^(/H-I " ff); b) corrector step

«r = « ? - ^ ( / r - / ; - i ) ;

(6.32)

< + 1 = ^«+<*)-

(6.33)

c) final step

Here /* = /(it*)- The MacCormack scheme is second-order accurate in both space and time (Hirsch 1990). For f = cu this scheme corresponds to the advection equation. 6.2.4

A stable

scheme

A stable scheme can be obtained replacing un in the fluxes in Eq. (6.15) by Then, we get an implicit scheme, viz. un+\ AXW+1

~ «?) = - A t t f & k - / ! % ) .

(6.34)

However, without additional refinement this scheme is not suitable for shock representation as it leads to a generation of noise and consequently to shock demolition (e. g., Oran and Boris 1988). Implicit methods are described in more details in Sec. 6.6. 6.2.5

Hermitian

compact

scheme

Hermitiem compact scheme can be obtained with the use of the relation which approximate the first-order derivative (Tenaud, Gamier, Sagaut 2000),

86

Numerical methods for a scalar hyperbolic equation

P ( / , x j _ 2 + J,xi+2) + aKJ,xi-i

°

+ f,xi+i)

+ f,xi —

( / i + 3 - /»-o, i - 3 ) +• 7 T r. (V/ *^ +^ 2 - / •<"-— 2 )' + ^ -Ax( / i + i - / i - i ) . 4 A a 2

6Aa.W1-a

(6-35)

For sixth-order accuracy, the constants must satisfy the constraints a=|, 6.2.6

Upwind

0 = 0,

o=y,

6=^,

c = 0.

(6.36)

differencing

Spatial (centered) discretization (6.13) in Eqs. (6.14) and (6.17) does not reflect the physical nature of the advection equation as it uses both information coming from the left and information coming from the right, without putting any priority on any direction. On the other hand, the following discretizations, though only first-order accurate, respects the physical nature of the equation 1/

n + 1

1

— 1/n

A, At

7/™ — II1}

+c

A * Ax

.

=0

(6-37) '

v

or u?+1-u? . u - ' + i - wL A, - + C"J±7 = 0(6-38) v At Ax ' The scheme of Eq. (6.37) will run stably on a computer only if cAt 0 < ^ < 1 .

(6.39)

As At and Ax are positive, this implies c > 0. On the other hand, the scheme (6.38) is stable provided cAt -1< — < 0 .

(6.40)

Hence, c < 0. So, it occurs that scheme (6.37) ((6.38)) is better in the case when c > 0 (c < 0) for which the fluid flows from left to right (right to left) and uf is updated based on the value u"_x (u" +1 ) to the left (right), in

Simple finite-difference schemes

87

the upwind direction. The above schemes are called upwind (donor-cell) methods. They are first-order accurate. The upwind scheme for the conservation equation, u,t + f,x - 0,

/ = cu,

(6-41)

/i+1/2

- 1 / 2 = 0,

(6.42)

can be written as tt t + '

A"

A

Ax

where: fi+l/2 - C+Ui + C~Ui+1,

= C + t*i_i + C~Ui,

fi-l/2

(6.43)

with c + = max(c,0) = - ( c + |c|),

c~ = min(c,0) = - ( c - |c|).

(6.44)

This scheme was generalized for a nonlinear conservation law with speed c = / „ by Godunov (1959) who used the Riemann problem (see Sec. 6.10.4). However, the scheme is only first-order accurate in space and therefore very diffusive and impractical in use. As a consequence of deficiencies of scheme (6.42) it is worthwhile to improve it by utilizing second-order upwinding in which a piecewise linear interpolation function for u(x, t) is applied within each numerical cell, i. e. IS.II7>

u(x, t) = < + (x - iAx) —J-, Ax

(6.45)

where: A < = uiX(iAx, tn)Ax.

(6.46)

The Fromm's scheme uses central differencing A< =

< + i

;

<

-

1

.

(6.47)

Despite of linear interpolation u(x, t) may exhibit jumps at the interfaces. It appears that upwind differencing stabilizes profiles (for |cAt/Ax| < 1) which are liable to undergo sudden changes. As a consequence of that the upwind differencing is perfectly suited to shocks and other discontinuities (e. g., Huynh 1995).

Numerical methods for a scalar hyperbolic

88

equation

Upwind methods were extended for vertex-based schemes for scalar advection equation and compared with finite-element methods, based on linear triangular elements, by Carette et al. (1995). Equation (6.37) ((6.38)) for c < 0 (c > 0) consists a downwind scheme. However, this scheme possesses the undesirable property that is unconditionally (for all time steps) unstable and therefore it is useless in practice. 6.2.7

Method of lines

discretization

In this method we first discretize the advection equation in space alone, which gives a large system of ordinary differential equations. Each component of this system corresponds to the solution at some grid point. This system can be treated with a use of a method for ordinary differential equations. For example, we might discretize the advection Eq. (2.12) in space at grid point Xi by Uj+i -

u* — c-

Uj-i

2Ax

'

(6.48)

i = 1,2,... ,m.

Now, Ui(t) is the solution along the line forward in time at the grid point Xi. This system can be written in the matrix form (6.49)

Uj t = Au + b,

where:

f° 1

1 0 1 1

c A = 2Aa;

•0

1

I

\

••

1

•• 0 1 1 O)

c , b= 2Ax

0 0

(6.50)

V um+i J

Here u0 and u m + i are the left and right boundary conditions, respectively. This approach is sometimes used in practise with an application of a software package for systems of ordinary differential equations. But the resulting method is often not as efficient as specially designed methods for the partial differential equation.

Temporal

6.3

discretization

89

Temporal discretization

We present here several methods for discretization of temporal derivatives in the advection equation, paying special attention to higher-order methods which can find potential applications due to there high accuracy.

6.3.1

Runge-Kutta

methods

Runge-Kutta methods are mainly appreciated for their high-order of accuracy. There are many Runge-Kutta schemes. We present first a third-order Runge-Kutta method for Eq. (2.16). This method was proposed by Shu and Osher (1988). It involves intermediate solutions u^ and u^ at the end of each stage, i. e. u*1*

=

un-Atf}X(un),

(6.51)

UW

=

^™ + i u ( i ) - l A t / , > « ) ,

(6.52)

un+1 = lu» + luW-ltoUul2)),

(6-53)

where At is the time step and ftX{um), m = 1,2, is evaluated at the intermediate state u = um. The other example is the fourth-order 4-stage Runge-Kutta method which is given by u (0)

=

-/,x(un),

(6-54)

„W

=

-f,x(u

n

+ ^«<°>),

(6.55)

u (2)

=

_/ia(un

+

u<3>

=

1

=

- / , x ( u n + A*u<2>), At, 2 (3) ). u " + f±i(u(o)+2u«+2u( >+u

u "+

^u(i)

) >

(6

.56)

(6.57) (6.58)

Runge-Kutta methods of arbitrary-order were presented by Oran and Boris (1988).

90

6.3.2

Numerical methods for a scalar hyperbolic

Multigrid

equation

methods

We consider the spatially discretized equation which can be written as follows:

§=/(«).

6 59

(- )

where f(u) denotes the discretization of spatial derivative terms. A multigrid scheme for this equation eliminates the high-frequency errors on the fine grid. As further fine grid iterations would result in a convergence rate (fractional reduction in error per iteration) degradation the solution is transferred to a coarser grid. On this grid, the low-frequency errors of the fine grid mansifest themselves as high-frequency errors, and so they are eliminated efficiently. This procedure can be applied recursively on a sequence of courser grids. To switch from one grid into another cycling strategies are developed. The most common cycling patterns are the V-cycle and the W-cycle. For instance, the V-cycle starts on the finest grid of the sequence, where one time-step is performed. The solution is then interpolated to the next coarser grid, where another time-step is performed. This procedure is repeated on each coarser grid until the coarsest grid of the sequence is reached. At this point, the coarse grid corrections are prolongated back to each succesively finer grid until the finest grid is reached. The combination of the V-cycles results in the full multigrid procedure (e. g., Wesseling 1991, Mavriplis, Venkatakrishnan 1995). Although the multigrid methods are very efficient they suffer few drawbacks one of which is that they require the construction of coarse grid levels for the solution of the fine grid equations, which is time consuming process.

6.3.3

Linear multi-step

methods

An s-step linear multi-step method has the following form: s

s

J2 aiun+i = At J2 hf{un+i) t=l

j=0

(6.60)

Finite-volume methods

91

with as — 1. For bs — 0 this method is explicit, otherwise it is implicit. As a special case we can distinguish the Adams method: s un+s

= un+s-l

bif(uH+i)

+^t^2

(6.61)

i=0

for which as = 1,

as-i

= —1,

cti — 0 for i < s — 1.

Another special case is the explicit Nystrom method: un+s

6.4

= un+s-2

s-l + A t ^ hf{un+i). i=0

(6.62)

Finite-volume methods

For physically motivated equations, it is important to insure that the numerical techniques conserve the physical quantities such as mass, momentum, charge, and energy. Consequently, it is often preferable to use a finite-volume method rather than a finite-difference method in which u™ is considered as an approximation to the average value of u(x, t) over a grid cell rather than a pointwise value of u. This average value is the integral of u over the cell divided by its length. A finite-volume scheme is based on the discrete equations which are constructed by expressing the integral conservation law on a discrete set of control volumes. The spatial divergence (derivative) terms are expressed as a surface integral of fluxes which are approximated with a use of solutions at two adjacent finite-volumes. The value u(x, t) will be approximated by its average value over the i-th numerical cell at the time tn = nAt, u? ~ - — / u(x,tn)dx, Ax i Jx{_1/a

(6.63)

where AXJ = xi+i/2 — ^i-1/2, the index i corresponds to the cell center and i - 1/2 (i + 1/2) to its left (right) interface. We consider Eq. (2.16). The integral form of this equation, applied to one numerical cell over a single time-step, leads to I •'Zi-1/2

[u(x, tn+1)

- u(x, tn)] dx =

92

Numerical methods for a scalar hyperbolic

I.

" + 1 [f(u(Xi_1/2,t))

- f(u(Xi_1/2

equation

+ Axht))]dt.

(6.64)

Hence, dividing by Ax; and using Eq. (6.63) we have un+1 = < where j^±xi2

IS a n

/"+1/2~^-1/2,

A

(6.65)

approximation to the average flux / ' " + 1 fWxi±1/2,t))dt.

75=1/2 - ^

(6.66)

As a special case we discuss the linear advection Eq. (2.12) which can be discretized with spatial central differencing, ^ - ^ - c A ^ - ^ -

1

.

(6.67)

Then, the flux / " is denned at the cell interfaces as In

_

Ji+l/2

-

C

" " + "i+1 2

In

'

_

• ' i - 1 / 2 ~~ C

*tf-l + Uj

((.(.R)

2

(,0-DOj

and the advection equation is discretized in the form of Eq. (6.65). The flux /j + i/2 can be chosen as /i+i/2 = | ( / ( n D + / « + 1 ) ) .

(6.69)

However, for such choice the above schemes of Eqs. (6.65) and (6.67) are unconditionally unstable. A much better choice is 1 Ar /<+i/2 = 2(/(«?) + /(«Cn)) - 2At("?+i - O

(6J°)

which leads to the Lax-Wendroff scheme (Lax, Wendroff 1964). It is noteworthy that as Xi is denned at cell centers, 2Aa;j differs from Xi+i — Xi-i. This can lead to low accuracy on irregular grids (e.
Von Neumann

stability

analysis

93

book unless otherwise stated. For simplicity reasons, however, we drop the bar symbol in u™ and use u" instead. There are also cell vertex schemes (e. g., Rudgyard 1993) which, on the contrary, have Ui defined not as the average over a cell < Zj_i/2,£i+i/2 >> but as the point value at Xi, and the representation U{ is continuous at X{. 6.5

Von Neumann stability analysis

In the numerical schemes, there are typically some restrictions on the relation between the time-step At and the spatial grid size Ax to make the numerical method stable. We explain this for advection Eq. (2.12) which is discretized by scheme (6.14) with a use of equally spaced points along both the t— and a;—axes. To check whether this scheme is the von Neumann stable we introduce a Fourier mode un

= AnejkiAx,

(6.71)

where j 2 = —1, fc is a wave vector and A = A(k) is called the amplification factor which is a complex function of wavenumber k. A difference equation is called stable in the von Neumann sense if \A(k)\ < 1

(6.72)

for some k. To find A{k) for Eq. (6.14), we substitute (6.71) back into (6.14). Dividing by An, we get rAt A{k) = 1 - j — sin (kAx).

(6.73)

Hence, we have that \A(k)\ > 1

(6.74)

for all k. This implies that discretization (6.14) is unconditionally unstable; there is no such At which would satisfy stability condition (6.72). It is noteworthy here that it is safer to run an algorithm with a lower time-step than the longest stable time-step. Various effects such as nonlinearity, variable grid and other parameters tend to enforce a practical stability limit which can be a factor of even two more stringent than the mathematical limit.

94

6.6

Numerical methods for a scalar hyperbolic

equation

Explicit and implicit time integrations

A problem, which is associated with a stability of a numerical scheme, is a choice between explicit or implicit methods. A disadvantage of explicit scheme is that the time-step At has to satisfy the CFL (Courant, Friedrich and Levy) condition which states that grid spacing cannot exceed the distance that the maximum characteristic speed in a physical problem can travel within the numerical time-step utilized, i. e.

AtcFL <

mini Ax) —-, Umax

.„ __. (6.75)

where vmax is the maximum of ail characteristic speeds. For problems which involve large variable gradients, in old schemes numerical viscosity was needed to achieve numerical stability. Thus, a numerical diffusion term in the form fiutXX, was added to the original equation; \i was called artificial viscosity in analogy with the physical viscosity of a nonideal fluid (von Neumann, Richtmeyer 1950). Another way to implement viscosity was provided in the Lax-Wendroff and Lax-Fredrichs schemes. See Eqs. (6.22) and (6.26). In these schemes /x = c 2 Ai/2 and /x = (Ax)2/(2At), respectively. By this way shocks were smeared to a width resolvable by the grid without significantly altering the overall flow (Jameson, Schmidt, Turkel 1981, Rizzi, Eriksson 1984). The introduction of numerical viscosity gives rise to another restriction on time-step. To obtain numerical stability of the central discretization of the numerical diffusion term fj,uiXX, the following criterion has to be satisfied: A*d

u

1 5- < - . min((Ai) 2 ) _ 4

. . (6.76) '

Thus, the strategy of determination of the appropriate time-step At is to choose between the two time-steps which are dictated by the CFL condition (AtcFh) and the numerical diffusion term (Atd), i- e. At = min(AtCFL,

Atd).

(6.77)

Explicit and implicit time

6.6.1

An implicit

integrations

95

scheme

We describe briefly here a fully implicit scheme that is based on a linearization of the equation u,t = / ( « ) ,

(6.78)

where f(u) is an operator representing the spatial discretization. One can integrate this equation with the trapezoidal method, obtained by averaging the two Euler methods: un+l

_

At

un

n = J[/(« ) + /(« n + 1 )]L

2

(6.79)

This scheme is a special case of an implicit one-step method such as u** 1 = un + At[6f(un+1)

+ (1 - 9)f(un)}.

(6.80)

The cases 0 = 0 and 9 = 1 correspond to the explicit and implicit schemes, respectively. To simplify the above discretization, we perform the following linearization: f(un+1)

~ / ( « " ) + f,u(un)(un+1

- un),

(6.81)

where fiU is the Jacobian. Consequently, we obtain an equation for the unknown un+1, viz. l-0At/,„(un)

V

'

The implicit method turned out to be inaffactive in its early implementation for magnetohydrodynamic (MHD) equations. Its use for plasma simulations of the solar corona of strong magnetic field demanded almost the same amount of time as in the case of the explicit method (Schnack et al. 1990). 6.6.2

Semi-implicit

method

The semi-implicit method was developed to remove the time-step constraint (e. g., Harned, Kerner 1985, 1986, Harned, Schnack 1986). In this method new terms are added to the original equations and some of these terms are treated implicitly.

96

Numerical methods for a scalar hyperbolic equation

We explain this idea by considering the wave equation u%u = c2u,xx.

(6.83)

A new term is subtracted from each side of this equation u
- clutXX,

(6.84)

where Co is a constant coefficient which is chosen from stability considerations. Equation (6.84) is time differenced as un+1 - c2{M)2u%1

= u" + Atu3 + c2(At)2unxx

- cg(At) V

M

.

(6.85)

The method is unconditionally stable when Co is sufficiently large, CQ > c. It is noteworthy that for one-dimensional case the semi-implicit method provides no advantages over an implicit method. In three-dimensional problems, however, the semi-implicit method offers advantages over an implicit method as the constant coefficient semi-implicit terms require no convolutions and a tri-diagonal solution is needed. The prons and cons of using implicit methods versus explicit methods are often balanced. A main problem with implicit algorithms is that they tend to damp strongly the high-frequency Fourier components that cannot be resolved adequately. The implicit method usually introduces gigantic matrix inversion or in the best case inversion of a tri-diagonal matrix. As a consequence of that duration of one temporal iteration is lower for explicit methods. An advantage of using the implicit method is a possibility to imply a large time-step, without generation numerical instabilities. The implicit method is often advisable since it leads to a useful result in a lower number of iterations and consequently the method is quicker and cheaper than the corresponding explicit method. However, the time-step should be smaller than the characteristic time of a given phenomenon. When the timestep is large (compared to explicit time-step constraints) in a fully implicit method, only the longest waves in the system are properly represented. For wave problems, the advantage of the implicit scheme is not always obvious, since the time-step is limited by the physical nature of the problem, namely, the resolution of wave structure. There should be at least few grid points within one wave length in order to resolve wave structure. It means that the time-step has to be at most small factor of the wave period. A larger time-step may lead to unphysical results. Although an implicit calculation

Numerical

errors

97

is stable for a broad range of wavelengths it does not mean it is accurate, even at long wavelengths. 6.7

Numerical errors

While describing a physical system one encounters a number of errors such as the errors of measurements, modeling, cut-off for truncation, and statistical. While the measurements errors result from limited abilities of the experimental methods, the modeling errors occur as a consequence of many assumptions and simplifications made at mathematical modeling of physical phenomena. For example, a nonlinear system is often represented by a linear one and this simplification is better for smaller amplitudes and for shorter times of the duration of the given phenomena. Modeling errors can also reveal theirself while discretizing equations in non-conservative form. For example, the inviscid Burgers equation, «,t + (u 2 ),* = 0,

(6.86)

can be discretized by the conservative method A z « + 1 - u?) = At ( « _ 2 ) 2 - K)2)

,

(6.87)

and by the non-conservative method Ax(u?+1

- < ) = 2 At «?(«?_! - < ) ,

(6.88)

which corresponds to the quasi-nonlinear form of the Burgers equation, utt + 2uutX - 0.

(6.89)

It occurs that the conservation properties have to be mirrored in the numerical method. It was shown by LeVeque (1997c) that the non-conservative method represents shock solution with the incorrect speed. While most physical fields are associated with continuous spectra, numerical simulations are based on discrete spectra. As a consequence of limited power (speed and memory) of computers, a continuous Fourier transform is replaced by its discrete counterpart, introducing cut-off errors. Another example of a cut-off error is provided by the Taylor series of Eq. (6.2), which consists of the three terms only. The cut-off error in the spatial derivative (see Eq. (6.5)) is proportional to Ax and consequently it

98

Numerical methods for a scalar hyperbolic

equation

can be reduced when a finer numerical grid or more terms in the Taylor series are applied. Cut-off errors are also associated with a finite number of computer bits. For example, | = 0.(3) a computer can represent by a finite number of 3, say, 0.3333. The statistical errors are associated with a limited precision of the computer. They are proportional to a ~ \fN, where a is the standard deviation and N is a number of the operations. A reduction of the cut-off errors by grid refinement leads to an increase of the numerical operations N and consequently to an increase of the statistical errors. However, if a small change of the initial conditions leads to large changes of the results, then the statistical errors are large. Numerical modeling of the differential equations introduces discretization and round-off errors. Discretization error is defined as the difference between the exact analytical solution of the partial differential equation and the exact (round-off free) solution of the corresponding difference equation. Round-off error is the numerical error that is introduced after a repetitive number of calculations. It consists of: diffusion, dispersion, nonlinear instability, and Gibbs error. If we denote by: A analytical solution of the partial differential equation, D exact solution of the difference equation, and N numerical solution from a computer, then discretization error = A — D, round - off error — N — D. The global error of a numerical scheme is the difference between the true and computed solution, A — N. The numerical scheme is called convergent if the global error goes to zero as the mesh is refined. The scheme is of order m if the global error is 0((At)m + (Ax)m) as At ->• 0 and Ax -> 0. Numerical dissipation (diffusion) results from a finite length of a numerical cell and is a consequence of the variable spreading over the whole numerical cell; a value which is represented by a point is spread over the whole cell. Numerical dissipation is also introduced by cut-off errors. As the numerical diffusion leads to damping of the solution, its magnitude is measured by the amplification factor, un+1 * = -TH-.

A

u

k

(6-90)

Numerical

errors

99

where u£ denotes the fc-th Fourier harmonic which is measured at t — nAt. In general, the amplification factor Ak is complex number which can be written as Ak=Ar+jAj

=

\A\e^k\

j 2= -l,

(6.91)

where (k) is the phase shift that is often associated with dispersive errors (see below). The damping of the A;-th harmonic corresponds to \Ak\ < 1. The case of \Ak\ > 1 is associated with numerical instabilities. For an ideal numerical algorithm \Ak\ should be equal to 1. As a consequence of nonlinear instabilities and the fact that a small magnitude of the diffusion can play a useful role, a safer choice is to set \Ak\ close but smaller to 1. Numerical errors which are associated with numerical diffusion are not usually high for large spatial scales which are of the order of several numerical cells. The amplification factor for such scales is close to one. On the other hand, the spatial scales of the order of one numerical cell length are quickly damped by diffusion. Dispersive errors occur when a phase speed i>^(fc) of the Fourier harmonic differs from the flow speed v. This type of error is a consequence of the approximation of spatial derivatives by finite-differences. Then, the phase speed is no longer independent of the wave vector k. An initial wave packet is a superposition of harmonics with different wave vectorsfc.Due to numerical dispersion this wave packet spreads in time. Dispersion is usually higher for shorter wavelengths and as a consequence numerical damping reduces the dispersive errors. Numerical dispersion is measured by a relative phase speed error P(fc)

=

H
(6.92)

It is noteworthy that since the phase errors grow secularly in time and the amplitude (diffusion) errors are self-eliminated, phase properties of the numerical scheme are more important than its errors associated with numerical diffusion. Gibbs errors result from discrete representation of a continuous function; the higher harmonics of the continuous solution are not included in the numerical representation. Evolution of this function in time reveals initially hidden oscillations (undershoots and overshoots) which exist between the grid points; they appear when a steep-sided profile moves a fraction of a cell from its initial location. These oscillations are worse near large

Numerical

100

methods for a scalar hyperbolic

equation

gradients and discontinuities. They may force appearance of the negative mass density. Elimination of these oscillations is not generally easy task. Nonlinear instabilities can be well explained with a use of inviscid Burgers Eq. (5.1). The nonlinear term uutX is responsible for energy transfer from long into short waves. As a consequence of this process a shock wave is generated. The shock wave is represented by short waves. As a linear stability of the numerical algorithm usually depends on wavelength, so the algorithm, which was stable for small amplitude wave (in linear regime), can be unstable for large amplitude wave (in nonlinear regime). The above discussion shows that hyperbolic equations cannot simply be discretized with a use of standard numerical schemes. Numerical errors which are encountered for such approaches reveal theirself by spurious modes and oscillations (overshoots and undershoots) at steep profiles. 6.7.1

Spurious

modes

A spurious mode is denned as an unphysical set of discrete values which satisfy the numerical scheme together with boundary conditions. We present a spurious mode for advection Eq. (2.12) for which the centered Euler scheme is applied to represent the spatial derivative. See Eq. (6.14). Equation (6.14) admits at steady state (d/dt = 0) apart from the physical mode u™ = uo = const the spurious mode

«? = U o (-l)'. This mode decouples the odd and even mesh points. It can be cured by physical constraints which couple the odd and even nodes by introducing a numerical diffusion, additional boundary conditions or direct filtering. Occurrence of spurious modes in numerical simulations is a consequence of dispersive errors. 6.7.2

Overshoots

and

undershoots

Overshoots and undershoots reveal theirself at steep profiles such as shocks or contact discontinuities as the discretization is not able to resolve short

Numerical

errors

101

oscillations (the smallest spatial scale which can be resolved on a mesh Ax has a length 2Ax). Consider the Cauchy problem for advection Eq. (2.12) with c > 0, viz. uo_[0,

»
-\l,

i>L

U i

(6.93)

According to Eq. (6.14) the spatial derivative utX > 0 at x = IAx. Then, uj < 0 there and therefore u/ will decrease in time, producing an unphysical undershoot at x = IAx. As a consequence of conservative properties of the scheme, this undershoot is usually accompanied by an overshoot which occurs at x = (/ + l)Ax. See Fig. 6.2. U

Fig. 6.2 Undershoot and overshoot at x = IAx and x = (I + l)Ax, respectively. These overshoots and undershoots can be eliminated by changing the spatial discretization to obtain a monotonic scheme. This can be done by applying locally a first-order scheme which smoothes overshoots and undershoots but does not contaminate regions of smooth profiles. 6.7.3

Monotonicity,

positivity,

and

causality

Monotonicity constraints is very important for development of smooth profiles in modern schemes. A grid function is called monotonic if the following condition is satisfied: min(ui-i,Ui+i)

< w» < maa:(ui_i,M»+i).

This condition prevents an occurrence of local maxima or minima at Uj.

102

Numerical methods for a scalar hyperbolic equation

A numerical scheme is monotonicity preserving if a monotonic function at t = nAt will be kept monotonic at t = [n + l)Ai. Monotonic schemes do not accentuate local maxima and local minima are non-decreasing. To measure the monotonicity total variation (TV) is defined as follows: oo

/

i

•OO

This measures the total amount of oscillation in the function u. Based on the total variation, we make the following definitions. A numerical method is called total variation stable (TV-stable) if there is a constant c depending only on the initial data such that TV{un)


(6.94)

for all n = 0,1, • • •, on all grids, no matter how fine. There is one way to guarantee TV-stability. That is to require that the total variation be nonincreasing as time evolves, so that the total variation at any time is uniformly bounded by the total variation of the initial data. This requirement gives rise to the very important class of total variation diminishing (TVD) methods. A numerical method is called TVD if TV(un+1)

< TV(un)

(6.95)

for all grid functions un (Harten 1983, 1989). TVD methods (e. g., Yee 1989, Tanaka 1994) are monotonicity preserving as they eliminate oscillations near high-gradient regions. This means that if the initial data is free of oscillations, then the solution should be monotonous for later times. The TVD criterion has led to the construction of many schemes for hydrodynamic (e. g., Yee 1989) and MHD equations (e. g., Tanaka 1994). The TVD concept as a measure of monotonicity does not readily extend to two and three dimensions. The definition of TV in two-dimensions, TV(u) = J2 ( K j - U i - i J + Kj

- Uij-il),

(6.96)

leads to only first-order schemes. As a consequence of that it is necessary to introduce a different measure of monotonicity. We discuss this for conservation law (2.16) which can be discretized as u

i,t = J2ciui+j 3

(6-97)

Numerical

errors

103

with

3

This condition follows from the fact that a constant solution should be preserved in time. With the use of (6.98) Eq. (6.97) can be rewritten as "*.* - ^2ci(Ui+i

~ "»)•

( 6 - 99 )

A scheme is called positive if Cj

> 0, j ^ 0.

(6.100)

From Eq. (6.98) it follows that for a positive scheme co = - ]T) Cj < 0.

(6.101)

Positivity ensures the following monotonicity property: If Ui is a local maximum, all terms of the r.h.s. of Eq. (6.99) are negative. Hence utt < 0 and u cannot increase at a; = X{. On the other hand, if «, is a local minimum, Ui cannot decrease. It has been noticed that for highly energetic flows, a numerical scheme can result in the kinetic energy of the fluid being higher than the total energy. This implies that the pressure is negative which is non-physical and the scheme is not positive in this case. It occurs that the upwind scheme of Godunov (1959) is positively conservative, whereas any Godunov-type scheme that is based on a linearized Riemann solution is not. However, Roe's method of Sec. 8.5.2 can be made positively conservative by appropriate replacement of the wave speeds (Einfeldt et al. 1991). This replacement also makes Roe's scheme entropy satisfying, a procedure which is only valid for the first-order method. For the entropy fix see Sec. 8.6. Causality requires that a fluid being convected from point A to B must pass through all the cells on a path between them. Causality and conservation are usually implemented by using locally computed fluxes to move quantities between adjacent cells. For instance, causality problem can be present in an implicit scheme which transmits information across a large distance using a large time-step At. If At is too large, the material in front of a shock has no forewarning this shock is coming.

104

Numerical methods for a scalar hyperbolic

equation

Reversibility means that the equation is invariant under the transformation t -> —i. As a consequence of numerical diffusion reversibility is not easy to achieve even when an algorithm is designed to be reversible. So, reversibility can be used as a measure of numerical diffusion. A check that can be made on the system is to run the calculations for many time steps until t = tn. At time t n the computation is stopped and the time-step is made negative. If by this way the solution retraces its evolution back to initial time the scheme is reversible. 6.8

Problems with source terms

There are various methods to deal with source terms. For example, in unsplit methods a single formula is used to advance the full equation over a time-step. In fractional step methods, the equation is broken down into pieces, and numerical methods are applied independently to every piece. 6.8.1

A fractional

step

method

We consider the following advection equation with the source term s: u,t + cutX = s.

(6.102)

A fractional step method for this equation is applied by first splitting it into two subequations u,t + cu,x

=

0,

(6.103)

u,t

=

8.

(6.104)

These equations can be solved independently by standard numerical methods by first solving Eq. (6.103) over At and then Eq. (6.104) over At too. Another possibility is to apply Strang splitting (Strang 1968). The idea of this method is to solve Eq. (6.103) over At/2. Then, we use the resulting solution as data for At on Eq. (6.104) and again Eq. (6.103) is solved over At/2. It is noteworthy that the fractional step method leads to inaccuracy in some cases, for example if the solution is nearly in steady state with cutX ~ s and we want to resolve small amplitude perturbations of this state. For instance, this method has to be refined for the gravity source

Open boundary conditions

105

term in the Godunov-type schemes, leading to numerically induced flow along the gravity. A cure of this can be made by the Godunov-splitting which is based on a solution of a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source terms (LeVeque 1997c). Another option is to apply the relaxation Riemann solver (LeVeque, Pelanti 2001). Source terms are often associated with physical phenomena which occur on faster time and smaller spatial scales than our time and spatial steps. Such source terms are called stiff and they require a special numerical treatment such as implicit methods. Examples of stiff source terms are provided by LeVeque (1997c).

6.9

Open boundary conditions

For wave propagation problems, open boundary conditions are often desired since the numerical box is restricted to a finite computational domain and a wave will propagate out of the box eventually. Incorrectly treated boundary conditions can lead to artificial reflection of the wave at the boundaries, which give non-physical results in the computational domain. Different methods have been investigated to treat correctly the so-called open boundary conditions. A simple method for imposing boundary conditions is to generate image cells at the boundaries of the simulation region and then, to compute the flux across these boundaries. This treatment is very easily implemented in an explicit scheme in which the fluid variables in the image cells are set at the beginning of each time-step. For implicit schemes, this approach can also be used although for large time-step a considerable time lag between the solutions and the boundary conditions can lead to numerical instabilities. A commonly applied method of setting open boundary conditions is known as the Sommerfeld condition. Among many versions of this method, the approach developed by Orlanski (1976) has been successfully applied to many wave problems. In this approach the wave propagation speeds at the boundaries of a numerical box are calculated by using variables values at the nearby interior grid points. The following is a detailed description of the method. Let u represents a physical quantity and x is the normal direction to the local numerical box boundary. The Sommerfeld radiation condition

Numerical methods for a scalar hyperbolic equation

106

is formulated as follows. Consider the scalar advection Eq. (2.12). The centered representation of this equation is given by

Here Ax is the local grid spacing, / represents the index of a right boundary grid point and superscript n denotes values at previous time-steps. From Eq. (6.105), replacing / by / — 1, we have

C=

-Atu?_1+ur?-2uU

(6 106)

-

and without this replacement Ul

\2At

+

2A^J - U"

2

\2At

2AxJ+U^Ax-

((U07)

Replacing n by n + 1 and rearranging the latter equation becomes i

u +1

i

yr

r

" ' = TT7

url +

TT7""-1'

(6J08)

where r = A l c . If the calculated c is higher than Ax/At, c will be set equal to Ax/At. If its negative sign indicates an incoming (into the simulation region) wave, c will be set equal to zero. For 1=1 corresponding to the left boundary, the outgoing wave is in the direction of index-decreasing. Eqs. (6.106) and (6.108) are then replaced by Ax v% - u%-2 * = T7.„ . ,V2 » n-i.

<+1 = i 3 7 u ? _ 1 - T^U"-

(6-109) (6 110)

-

It is noteworthy that implementation of the Sommerfeld radiation condition in the Godunov-type schemes is not required as these schemes are based on the method of characteristics which essentially does not produce any reflection at open boundaries.

Shock-capturing schemes

6.10

107

Shock-capturing schemes

Shock-capturing schemes can be divided into geometric approaches and algebraic approaches (e. g., Murawski, Tanaka 1997). While all of the members of the geometric approaches attempt to assine a value to the generalized density, u, in a global manner the algebraic approaches use flux limiters. The algebraic schemes enforce some constraint on the problem, usually that some component of the solution be total variation diminishing (e. g., Yee 1989). 6.10.1

Algebraic

schemes

In the schemes which are based on the algebraic approach the flux can be represented by fi+l/2 = Cj+l/2/j+i/2 + (1 L

C

i+l/2)fi+l/2i

(6.111)

H

where f and f are low-order and high-order approximations to / , respectively. These fluxes are associated with the Godunov and Lax-Wendroff schemes of Eqs. (6.113) and (6.22), respectively. The weight c is usually bounded by 0 and 1, and is computed through the use of flux limiters. Some common limiters are: minmod, superbee, van Leer, and monotonized centered. Minmod is the most diffusive limiter of the above, in the sense that it adds less downwind contribution. This limiter selects the wave with the smallest norm of the two compared, provided the jumps across the waves are in the same direction, i. e. the wave strengths have common sign. If not, the wave is entirely suppressed. On the other hand, the superbee limiter is known to be overcompressive, i. e. it tends to sharpen profiles into discontinuities. The monotonized centered limiter seems to be a good choice in most situations as it is less dissipative. The sufficient condition for the TVD stability of Eq. (6.111) is q + i ^ - ^ ^ ^ X ) . '

Ui-

(6.112)

Ui-i

This condition is satisfied by most of common limiters. These flux limiting schemes can be divided into two types: the flux corrected transport (FCT) method (Book 1981, Boris, Book 1976) and flux limited schemes. The best-known member of the latter family is the (/>-limiter scheme of Sweby (1984).

108

6.10.2

Numerical methods for a scalar hyperbolic

Geometric

equation

schemes

The geometric schemes are: the Godunov scheme (Godunov 1959), the monotonic upwind scheme for conservation laws (MUSCL) scheme (van Leer 1979), piecewise parabolic method (PPM) scheme (e. g., Dai, Woodward 1994a,b, 1996), and essentially non-oscillatory (ENO) scheme (Harten 1989, Casper, Atkins 1993, Liu, Osher, Chan 1994). In the geometric schemes the defining functions can be continuous within a cell, but may be discontinuous at cell edges. 6.10.3

Godunov

method

In the original Godunov approach, the upwind finite-volume flow solver was used. The solution was considered to be piecewise constant over each grid cell at a fixed time, i. e. U(x) = Ui, £i_i/2 < X < Xi+i/2.

(6.113)

So, discontinuities are placed at the cell interfaces £,±1/2- The evolution of the flow to the next time-step results from the wave interactions originating at adjacent cell boundaries, specifically Riemann problem (e. g., Dai, Woodward 1994a). As it mimics much of the relevant physics, Godunov scheme would result in an accurate and well-behaved treatment of shock waves. However, this approach leads to first-order accuracy. As a consequence of that the Godunov scheme exhibits strong numerical dissipation, and discontinuities in the solution are considerably smeared over several grid zones. For comparison, the Lax-Wendroff scheme of Eq. (6.22) is more accurate in smooth parts of the solution. But near discontinuities, numerical dispersion generates oscillations, also reducing accuracy. The Godunov method is based on solving the Riemann problem. 6.10.4

Riemann

problem

The Riemann problem corresponds to the initial condition which represents two semi-infinite states

for Eq. (2.16). Here, ui and uT are constants which denote to the left and right states, respectively. The solution to the Riemann problem represents

Shock-capturing schemes

109

a wave that propagates partly leftwards and partly rightwards. It is noteworthy that a sonic point uo for the nonlinear advection equation uj + f(u)tX — 0 is such that /,„(uo) — 0. Then, the value ito propagates with its speed 0. This point is also called the stagnation point.

6.10.5

The MUSCL

scheme

The low accuracy and the complexity of the Godunov method meant that other methods needed to be devised. Two decades passed until the Godunov approach was first extended to second-order spatial accuracy by the MUSCL approach which was developed by van Leer (1979). His approach consists of two key steps: an interpolation (projection or reconstruction) step where, within each cell, the data is approximated by linear functions, and an upwind step where the average fluxes at each interface are evaluated by taking into account the wind direction. A great deal of effort was spent to enhance the accuracy of the interpolation step, and to improve the efficiency and robustness of the upwind step (Roe 1981, 1982). Accurate interpolations are derived by assuming that the data is smooth. However, in the presence of a shock, these interpolations lead to oscillations which can be prevented by an introduction of a monotonicity constraint for a numerical scheme (van Leer 1979). In this scheme the accuracy was increased by constructing a piecewise linear approximation of u(x, t) at the beginning of each time-step, viz. U(x,t) -Ui + Si(x-Xi),

Xj-l/2 < X < Zj+l/2,

(6.115)

where Sj is a slope and Xi = (xi + Xi+i)/2 = x» + Arc/2 is the center of the grid cell. So, u(xi,t) = u*. Moreover, it is required that the average value of u(x,t) over the cell is equal to U;. The slope Si can be constructed by many ways such as Ui+l Si

=

Ui-i

2Ax Uj -

Uj-l

Uj+i - Uj

Si

=

Ax minmod(

(centered slope, Fromm's scheme),

(6.116)

(upwind slope, Beam — Warming scheme), (6.117) (downwind slope, Lax — Wendroff scheme),(6.118) l

— , -^—r

-)

(minmod slope),

(6.119)

Numerical methods for a scalar hyperbolic equation

110

where the minmod function is defined by

{

a, b,

for \a\ < \b\ and ab > 0, for \a\ > \b\ and ab > 0,

(6.120)

0, for ab < 0. So, the minmod function returns the smallest argument in magnitude if the arguments are of the same sign, and zero if they are not. Choosing Si = 0 in the above expressions leads to the Godunov method. With the piecewise linear reconstruction, advection Eq. (2.12) with c > 0, is discretized as follows: < + 1 =«? - ^

-<-i) - \^(Ax

«

-

cA

*)(*r - s " - i ) -

(6-121)

This expression can be obtained from the flux-differencing <

+ 1

=<-^(/r

+

i-/D,

(6-122)

where: fr = ^-tJn+1cu(xi-1,t)dt.

(6.123)

Hence with the use of Eq. (6.115) and the relation u(x,tn+i)

= u(x -

cAt,tn)

we have

c < - i + ^(Aa; - cAi)s7_ 1;

c > 0.

(6.124)

For c < 0 we can show that /r = c<-f(Aa: + cAtK,

c < 0.

(6.125)

Fluxes (6.124) and (6.125) can be written in the compact form ft = c-un{ + c + < _ x + \\c\ ( l -

^

) s?Ax,

where: c~ = min(c,0),

c+ =

max(c,0).

(6.126)

Shock-capturing

schemes

111

In regions where Sj = 1, the reconstruction used is linear, and the truncation error is C((Ax) 2 ). In regions where s, = 0, the reconstruction used is piecewise constant, and the truncation error is O(Ax). The slopes Si are limited to enforce the monotonicity of the reconstruction.

6.10.6

Higher-order

schemes

A PPM scheme is an extension of the MUSCL scheme. The key difference from MUSCL is that u is allowed to be piecewise parabolic within a cell, rather than piecewise linear. Second-order accuracy in time is again achieved in the same way as in MUSCL, via characteristic tracing and solving Riemann problems. The PPM scheme was successfully applied to ideal MHD equations by Dai and Woodward (1994a,b). Essentially non-oscillatory (ENO) schemes are again extensions of the basic Godunov approach. Arbitrarily high-order polynomials are allowed to define u within a cell, yielding arbitrarily high-order spatial accuracy (Harten 1989). Various shock-capturing schemes were compared by Woodward and Colella (1984) by computing a blast wave interaction problem in one dimension. The result of that test was an ordering of the schemes in terms of the accuracy. With the most accurate schemes listed first, that ordering was as follows: PPM, MUSCL, and the Godunov scheme. A number of numerical schemes for advection Eq. (2.12) were tested by Odman and Russell (1998). These schemes included finite-element, finitevolume, pseudospectral, and advanced finite-difference methods which lead to similar results in most test cases. It is worth mentioning here that there are also front tracking methods (e. g., Hilditch, Colella 1995, LeVeque, Shyue 1995, 1996), where the tracked front is represented locally by a polygonal line which divides the numerical cells into two pieces. In each piece, the solution is updated by a method that is necessarily unsplit, in order to preserve the Rankine-Hugoniot conditions of Sec. 8.4 for the tracked front. The borders of the numerical domain are moved such that they coincide with the discontinuity. When multiple interacting discontinuities appear shock tracking methods become very intricate.

Numerical methods for a scalar hyperbolic

112

6.10.7

Kurganov

equation

schemes

As the above shock-capturing schemes are relatively expensive other secondorder schemes were developed (Kurganov, Tadmor 2000, Kurganov et al. 2001). In these schemes spatial discretization is robust for high gradients uiX despite of the fact that no characteristic decompositions or Riemann solvers were applied. These gradients are limited: u,Xi — minmod[a(ui+i

- Uj),a(u; - u,_i),

'+1 — — ] .

(6.127)

For a = 1 the minmod limiter is recovered. The monotonized central limiter corresponds to a = 2. The interface values are evaluated by u

\+i/2

=

Ui+2U>*v

(6.128)

<+i/2

=

«i+i - 2U-**+i'

(6.129)

where the superscripts I and r correspond to the left and right values to the interface xi+i/2. The Kurganov-Tadmor (2000) flux is

/i+i/a = j LfK+i/a) + /("in-i/a) + O i + i / 2 « n / 2 - «i+i/a)]>

(6-130)

where flj+i/2 = max(|t4 + 1 / 2 + c- + 1 / 2 |, K + i / 2 + 4+1/21)

(6.131)

and c is the sound speed. The Kurganov-Noelle-Petrova (2001) flux is

,

_

a 7 t +l/2/K+l/2)

Ji+1/2 —

+ <+l/2/K+l/2)

T —+ fl i+l/2 + °j+l/2 a a

t+l/2ai+l/2

^-«+i/2-«i+i/2),

i + l / 2 T+ l1a i + l / 2 "t+l/2

(6-132)

where a

i+i/2

=

-min(0,uli+1/2-

c\+1/2,uri+1/2

- cri+1/2),

(6.133)

Flux-corrected transport method a

t+l/2

6.11

=

maa: 0 U

( ' i + l / 2 + C '+l/2.<+l/2 + C i+l/2)-

113

(6.134)

Flux-corrected transport method

A flux-corrected transport (FCT) method is a modification of a finitedifference method which belongs to the family of algebraic approaches of Sec. 6.10.1. A FCT technique was originally developed by Boris and Book (1973a,b) for a simple conservation equation. This method was subsequently improved and extended to different geometries and more complicated cases (e. g., Book, Boris, Hain 1975, Zalesak 1979, Book 1981, Patnaik et al. 1987, Steinle, Morrow 1989, DeVore 1991). In particular, applications of the FCT method to general fluid systems, multi-dimensions, and curvilinear geometry were made by Book, Boris and Hain (1975) and Book (1981). A new algorithm for the implementation of a flux limiter in multi-dimensions was introduced by Zalesak (1979). This new flux limiter performs better than the old flux limiter of Boris and Book (1973a,b). For the MHD equations the accuracy of the method was improved by DeVore (1991) by placing the magnetic field components at the cell interfaces so that the magnetic field is guaranteed to be divergence-free. The aim of a FCT method is to minimize numerical errors in numerical algorithms. These errors may be crucial for an adequate representation of the solution. A simple finite-difference scheme can lead to numerically induced oscillations of the function u(x), which occur in a neighborhood of large gradients and are a consequence of dispersion and the Gibbs phenomenon. See Sec. 6.7. A FCT method filters these oscillations by introducing diffusion. In the neighborhood of extreme of the function u(x), the FCT method leaves some diffusion which eventually replaces one-point extremum by a few-points plateau. This excess of diffusion is removed by introducing anti-diffusive fluxes that are equal and opposite to the dispersive and Gibbs errors. As a consequence of that the FCT method is about one-order of magnitude more accurate than other simple finite-difference schemes. Beside that the FCT method eliminates nonlinear instabilities which reveal theirself at long numerical sessions with a use of standard schemes. Although the method is accurate for large gradients, it is less accurate for periodic functions than the fast Fourier transform method of Sec. 6.12.1. An advantage of this method over the fast Fourier transform scheme is an easy way of the im-

Numerical methods for a scalar hyperbolic equation

114

plementation of the boundary conditions. The FCT method is also quite universal since it is easily applicable for complex systems. However, a number of problems remain such as staircasing (Woodward, Collela 1984). The main aim of the present section is to assess the applicability of the FCT method to the conservation equation uit + f,x = s,

(6.135)

where the general density u, flux / = uv, v is the flow velocity, and the source term s are functions of x and t. This simplification provides us with the necessary computational and physical insight to complex problems. It is convenient to view a numerical method for solving Eq. (6.135) as the superposition of convection, diffusion and anti-diffusion. In the convection stage, the general density u is advanced from time t to t + At by using the finite-difference flux form for the spatial derivatives and the predictorcorrector method for the time integration. At this stage the source term s(x, t) is added to the transported equation. This simple numerical scheme introduces ripples into the solution. In the diffusion stage a strong numerical diffusion is added to the transported solution to remove the ripples from the solution. This numerical diffusion is subsequently compensated by introducing anti-diffusive fluxes that are additionally corrected. These corrected fluxes should get rid of numerical ripples. In what follows we give details of the different steps of the numerical scheme. 6.11.1

Convection

We use the predictor-corrector method (e. g., Harned, Schnack 1986) for time integration. The predictor-corrector method first predicts a solution u* by advecting Eq. (6.135) from time t to time t* = t + At/2, i. e.

AxiU* = AxiU? - ^(/f +1/2 - /f_1/2) + YAX^-

(6-136)

The resulting value is used to construct advective velocities and sources at t*. These advective velocities and sources are then used to update u from t to t + At, Axtuf

= AxiU? - At(f*+1/2 - /*_ 1 / 2 ) + AxiAts*.

(6.137)

Here the superscripts n and T denote old and transported solutions, respectively. At the cell's center there is a grid point with the coordinate

Flux-corrected transport

method

115

that is denoted by x = iAx and all the physical variables with the integer subscripts are the cell centered quantities. The grid is staggered as the flow velocity and consequently flux / is evaluated at the cell interfaces. The flux /i+i/2 is defined at the right interface xi+i/2. Note that the time scheme is centered. Such approach guarantees that the predictor-corrector method is accurate to the second-order in the time step (numerical errors introduced by the time integration are proportional to (At)2); the finite-difference flux form is accurate to the second-order in the spatial step Ax. 6.11.2

Diffusion

At this stage the transported solution uT given by Eq. (6.137) is diffused by introducing numerical diffusion to reduce numerically induced oscillations. We write

AxiUJD = Axiuf + (/£ 1/2 - /£ 1 / 2 ).

(6.138)

The diffusive flux is defined as / £ i / 2 = ^ + 1 / 2 A a : i + 1 2 + A a : i « + i - «?),

(6-139)

with the diffusion coefficient v (Boris, Book 1976),

vi+1+ViAt (

ei+1/2 = ^ —

T

1

( ^

+

1

\

A^7j'

miA-.y,

(6 141)

'

where v is the flow velocity. 6.11.3

Anti-diffusion

The transported and diffused density uTD contains an excess of numerical diffusion which has to be compensated to get an adequate solution. This is done by introducing the anti-diffusion flux xU x Axi+i + Axi T T /i+i/2 = Mi+1/2 5 («<+i - «* ),

,...„> (6.142)

116

Numerical methods for a scalar hyperbolic

equation

with the anti-diffusion coefficient (Boris, Book 1976) ^+1/2 = ^ ( 1 - ^ + 1 / 2 ) -

(6-143)

This uncorrected flux usually possesses unphysical extreme which can be removed with the aid of a flux limiter. In this case, Zalesak's flux limiter Z (Zalesak 1979) is used to obtain the corrected flux

&1/2

= Zi+x/2fY+l/2.

(6.144)

Finally, this corrected flux is applied to get a more accurate solution u " + 1 , i. e.

AxiU?+1 = LXiuJD - ( / 4 1 / 2 - f?_1/2). 6.12

(6.145)

Spectral methods

Spectral methods have been used in computational fluid dynamics since the early 1970's. These methods are continuously being refined and as a consequence of their high accuracy in comparison to common finite-difference methods they are the favorite numerical tool. A thorough description of the use of spectral methods in fluid dynamics is provided by Coutsias et al. (1989). In spectral methods, u(x, i) is expanded by a finite-number of orthogonal basis functions fo(x,t), fi(x,t), •••, fN-i{x,t) such as Fourier, Bessel, Chebyshev, Legendre series, JV-l

«(»,*)= ^

Oi(t)/i(af).

(6.146)

i=0

The set of coefficients {a*} is used for a representation of u(x). The basis functions are chosen to have properties that simplify calculations. For instance, these functions should be analytic and orthogonal as well as reveal the physical processes in the problem. Differentiation in configuration space then becomes an algebraic operation in transformation space. There are three basic approximations used in spectral methods: Galerkin, tau, and collocation. We discuss them for the equation utt — Ru,

(6.147)

Spectral methods

117

where R is generally a nonlinear operator. The initial and boundary conditions are: u(x, t = 0) = u0{x),

B±u = 0,

(6.148)

where B± are linear operators acting at the left (xmin) and right (xmax) boundary. If in expansion (6.146) the basis functions are assumed orthonormal and satisfy the boundary conditions then the coefficients {a*} are such that ai(t)=

fi(x)u(x,t)dx.

(6.149)

JXmin

Prom Eq. (6.147) we obtain N l

~

rxma*

°M = 5Z / n=0

fi(x)Rfn(x)dx.

(6.150)

*">•"

This equation together with expansion (6.146) are the Galerkin approximation (Fletcher 1984) for Eq. (6.147). In the tau approximation (Lanczos 1956) the basis functions in Eq. (6.146) do not have to satisfy the boundary conditions. Instead, there are two additional constraints, JV-l

J2 anB±fn(x) - 0.

(6.151)

n=0

In collocation methods, N grid points are chosen such that Xmin < Xo < Xi < • • • < IjV-1 < Xmax.

(6.152)

The expansion coefficients are then given by JV-l

X I anfn(xm)

= u(xm),

171 - 0 , 1 , • • •, N - 1.

(6.153)

71=0

Among spectral methods the most popular is the fast Fourier transform method and Chebyshev expansion.

118

6.12.1

Numerical methods for a scalar hyperbolic equation

The Fourier transform

method

The idea of the Fourier method is as follows. In most cases, the periodic function u(x) is represented by its values at the points Xi, i = 0,1, • • •, such that Ui = u(xi). Let Ax denotes a width of the interval between two neighboring points x\ and Xj+i. A reciprocal of 2 Ax is known as Nyquist's critical wave vector

kc = ^ - .

(6.154)

This critical wave vector is important as a result of the two connected but different reasons. First, if a continuous function u(x), which is represented by u,, possesses its spectrum band lower than kc, viz. !Fu(x) = 0 (where T is the Fourier operator) for |fc| > kc, we say then that the function u(x) is entirely described by its values u,. Second, if the function u(x) has no spectrum limited to values lower than kc, then the energy, which corresponds to k outside the interval — kc < k < kc is erroneously transformed to this interval. This phenomenon is called aliasing (e. g., Fornberg, Whitham 1978). The energy transfer is a consequence of the discrete Fourier representation of a continuous spectrum. Unfortunately after choosing such representation, a chance of removing the errors associated with the aliasing is rather low. One of the ways to remove these errors is to know a natural width of the spectrum. Then, we can choose a representation with properly small Ax in order to have large values of kc. A practical way of fighting with the aliasing is to check whether values of the Fourier transforms are negligibly small while \k\ -¥ kc. If these values are finite it is possible that k from outside the interval < —kc,kc > will be transformed into this interval, leading to erroneous representation of the spectral derivatives of the function u(x). Reduction of Ax will cause enlargement of the interval and in a consequence of that a reduction of the errors. In the past the aliasing consisted a serious numerical problem (e. g., Fornberg, Whitham 1978). However, the spectral Galerkin (Fletcher 1984) and tau (Lanczos 1956) methods are free of aliasing errors. They eliminate aliasing by removing spurious frequencies from the configuration space. As an example, we consider a continuous, periodic function u(x) in the domain 0 < x < 2n. The Fourier expansion of u{x) can be written

119

Spectral methods symbolically as oo

u(x) = a0 + Y,

a

mejmx

+ c.c. = ^ ( { a ™ } ) ,

(6.155)

m=l

u(x)e-jmxdx,

am = -!- /

(6.156)

2TT J0

where c.c. stands for the complex conjugate of the proceeding term. As this expansion requires infinite number of coefficients its numerical counterpart is N-l

u(xi) ~ UN(xi) = aQ + ^2 ame'mXi

+ c.c,

(6.157)

m=l

where X{ is the collocation point. For example, in the case of Fast Fourier Transform (FFT) method the collocation points are, Xi = —i,

i = 0,l,---,N-l.

(6.158)

The derivative is performed now with the a use of the following formula: N-l

utX ~ MAT,, = Y^ jmamejmx

+ c.c = ^" _1 0'm^"(u)).

(6.159)

m=0

It appears that for the fast Fourier transform the number of multiplications can be equal to NlogzN (e. g., Press et al. 1992), where N denotes the number of Fourier modes applied. The Fourier method is very accurate for periodic functions. It works also well for local functions such as the Gauss' function e~x . A disadvantage of this method is that it requires implementation of periodic boundary conditions. Both periodic and non-periodic boundary conditions are easily implemented by finite-difference and Chebyshev expansion methods. 6.12.2

The Chebyshev

expansion

method

For non-periodic boundary conditions the Fourier method cannot be directly applied. Instead, the expansion in Chebyshev polynomials can be used, Tn{x) = cos(n arccos x),

\x\ < 1.

(6.160)

120

Numerical methods for a scalar hyperbolic

equation

The first two Chebyshev polynomials are: 2o(a0 = l,

T1{x)=x.

(6.161)

Higher-order polynomials can be determined from the recursion formula Tn+1(x)=2xTn(x)-Tn_1(x),

n = l,2,---.

(6.162)

At the boundaries \x\ = 1 we have T B (±1) = ( ± l ) n .

(6.163)

The Chebyshev polynomials satisfy the following differentiation formula (e. g., Press et al. 1992): 2T n 0r) = ^ f + ^ f . n +1 n—1

(6.164)

If u{x) is expanded with a use of the Chebyshev polynomials, oo u x

( ) = ^2 amTm{x),

(6.165)

m=0

then 1

1

1 °°

xu(x) = -ax - (a0 + -a2)Ti(x)

+- ^

(a m _i + am+1)Tm(x),

(6.166)

m=2 oo

oo

/

u, s = J2 (2m + l ) a 2 m + 1 + 2 ^ 1 TTI=0

oo

^

n=l \ m=n+l,m+n=odd

\

m a m J T„(x).(6.167) /

A drawback of the Chebyshev expansion method is that collocation points are dense at the edges of a simulation region while leaving the central part of the simulation region less populated. In summary, as long as there are no sharp gradients, spectral methods require less resolution than finite-difference methods. The spectral methods are sometime superior to finite-difference methods although they do not guarantee positivity in the way finite-difference schemes do. However, for complex flows which contain discontinuities and shocks, spectral methods are inferior to finite-difference methods.

Finite-element

6.13

method

121

Finite-element method

There are two approaches adopted by finite-element method (FEM): global and local. The global (locall) approach is based on an expansion of u(x) over the whole domain (locally) with the use of Eq. (6.146). The basis (shape) functions can be chosen, for instance, as follows:

{

f0r X

*-l <x<xii iovXi<x<xi+u

l~-x~\ J ^

(6-168)

0 otherwise. So, each f%{x) consists of two straight lines, with a positive (negative) slope on the left (right). The region between two neighboring grid points is called an element. Other choices of fi(x) are also possible (e. g., Zienkiewicz, Morgan 1983). Generally, in the global approach each element is expanded in a polynomial /iW=^ai,i!,

(6.169)

i=i

where the coefficient an refers to the l-tb. term of the polynomial for element i. Now for the nonlinear advection equation we define an error function Ru such that Ru({fi},t)

= u,t + f,uu,x.

(6.170)

Ideally, Ru should be equal to zero. Then, we require f """ WmRu dx = 0,

m = 0,1, • • •, N - 1,

(6.171)

where {W m } is a set of weighting functions. Using expressions (6.168) and (6.169) in expansion (6.146) and then in Eq. (6.170), from Eq. (6.171) we obtain a set of coupled ordinary differential equations A a , ( + B a + D = 0,

(6.172)

where A, B , and D are matrices which are integrals over the entire region or over a subregion for global and local functions, respectively. The functions {W m } can be chosen in a number of ways. For instance, the choice Wm — fi

122

Numerical methods for a scalar hyperbolic

equation

leads to a Galerkin method and the choice Wm = S(x — xm) gives a point collocation method. 6.14

The locally one-dimensional method

The above discussed methods can be applied to solve multi-dimensional equations. For instance, instead of solving the two-dimensional equation « , t + /(«).*+ff(«),» = 0

(6.173)

we replace it by a sequence of two steps, each of which is performed in only one spatial direction. As an example we present the locally one-dimensional method:

ufc

=

«S- + At/(«5), a ,

(6.174)

«£"

=

U£+Ats(«:,.),v

(6.175)

These equations can be solved independently with a use of one of the methods which were originally developed for a one-dimensional conservation equation. The locally one-dimensional method is a one-dimensional version of the operator splitting method which is discussed in Sec. 9.1.

Chapter 7

Review of numerical methods for model wave equations

As it is already known from the former sections, many model equations for uni-directional long waves can be written schematically as

where the term D corresponds to dissipation or dispersion. For small time this term can be neglected and the most numerical schemes, which were presented in Chapter 6, can be adopted for solving equation (7.1). However, specific methods were developed for model wave equations. We briefly mention some of them here. A finite-difference scheme was applied by Zabusky and Kruskal (1965) to solve the Cauchy problem for the KdV equation. Results of the numerical experiments revealed that solitary waves do not loose their identities during their interaction with one another. Instead, these waves reappear virtually unaffected in size and shape. These properties are characteristic features of particle-like behaviour and they constituted a ground for coining the term soliton. Several numerical schemes such as the Fourier method and the finitedifference method were developed and compared for the KdV equation by Abe and Inoue (1980). The results of this comparison revealed that the Fourier method is the most accurate and effective. Few numerical boundary-value algorithms for computation of nonlinear waves and solitons were reviewed by Boyd (1990). Spectral methods for spatial discretization and Crank-Nicolson differencing scheme to advance the solution in time were developed for the KdV and KP equations by Wineberg et al. (1991). An algorithm was presented 123

124

Review of numerical methods for model wave

equations

by Sloan (1991) for the Fourier pseudo-spectral solution of RLW Eq. (4.93). The generalized KdV equation, U}t

<" " " , X

"I" i^iXXX

I ^,XXXXX

=

^*)

V

/

was solved by a mixed Chebyshev/radiation function pseudo-spectral method by Boyd (1991). This equation models capillary-gravity waves when the Bond number, which measures the relative importance of surface tension and gravity, is close to and slightly less than 1/3 (Hunter, Scheurle 1988). It is noteworthy that Eq. (7.2) possesses the solutions in the form of a weakly non-local soliton which consists of a central core, which resembles a classical soliton, accompanied by oscillatory wings which extend indefinitely from the core. An implicit finite-difference scheme was used by Delfour, Forting, and Payre (1981) to solve the multi-dimensional modified NS equation jutt + P\u\pu - V 2 u = 0,

j 2= -l.

(7.3)

The main feature of the scheme was that it satisfied a discrete analog of conservation laws of this equation. The numerical results exhibited in particular the formation of one-dimensional solitons. An efficient pseudo-spectral Fourier scheme for the cubic-quintic NS equation, ju,t + u,xx + Pc\u\2u + /3,|u| 4 u = 0,

(7.4)

was developed by Cloot, Herbst, and Weideman (1990). This equation arises when higher-order nonlinear effects are taken into consideration in comparison to the cubic NS equation which corresponds to Pq = 0. A generalized NS equation, ju,t+utXX

+ pc\u\\

+ Pq\u\4u + jPm\u\*xu + jpu\u\2utX

=0,

(7.5)

was solved by pseudo-spectral methods by Pathria and Morris (1990). This equation contains, as special cases, such equations as the cubic NS equation, the derivative NS equation and the cubic-quintic NS equation. It is known that in a homogeneous, one-dimensional plasma with a uniform magnetic field, a circularly polarized, transverse Alfven wave obeys the derivative NS equation (e. g., Spangler 1986) jutt + P{u\u\2)tX + autXX = 0.

(7.6)

125

A computer program was developed by Spangler (1986) to solve the initialvalue problem for this equation via the Fourier method. Numerical simulations of the Leibovich-Pritchard-Roberts equation (Leibovich 1970, Pritchard 1970, Roberts 1985a, Bogdan, Lerche 1988) «,t + w,x + uutX + RuiXXX = 0,

(7.7)

were performed by Weisshaar (1989) to show that its solitary waves interact like solitons. Here M

./-co , / ! + ( * - { ) '

A spectral method was used by Pelloni and Dougalis (2000) to develop numerical schemes for a class of nonlocal, nonlinear, dispersive wave equations such as the Benjamin-Ono (BO) (Benjamin 1967, Ono 1975) and the intermediate long wave (ILW) (Joseph 1977, Abdelouhab et al. 1989) equations. The BO equation was solved numerically by Miloh et al. (1993). The ILW equation models the long internal waves in an incompressible, stratified fluid, whose density varies in a thin layer which is located between a heavier and a lighter fluid. The equation is ut -\

r— ux + uu,x + KutXX = 0, o where K is the integral operator 1 r+°° TT(X - f} Kf{x) = -Y6Pj_ coth-^y-Hm-

(7.9)

(7-10)

Here 8 > 0 corresponds to thickness of the lighter fluid and P stands for a principal value.

Chapter 8

Numerical schemes for a system of one-dimensional hyperbolic equations

8.1

Linear system of one-dimensional equations

Consider the linear system of equations uit+Au,x=0,

(8.1)

u(x,t = 0) = u 0 (a;).

(8.2)

with the initial condition

Now, both u and A are matrices such as u : R x R -» Rq and A e Rqxq. The above system is called hyperbolic if a constant matrix A is diagonalizable with real eigenvalues and the corresponding set of right eigenvectors is complete (Jeffrey, Taniuti 1964). If these eigenvalues are distinct for all u(x, t) the system is called strictly hyperbolic. That allows us to decompose the matrix A in the following way: A = RAR-\ where: R=(r1|,r2|,---,|r«) is the matrix of right eigenvectors and A = diaff(A\AV--,A«) is a diagonal matrix of eigenvalues. 127

(8.3)

128

Numerical schemes for a system of one-dimensional

hyperbolic

equations

Prom Eq. (8.3) we get A R = RA. Hence Arm=Amrro,

m = 1,2,-•• ,q.

(8.4)

From this equation we can obtain the eigenvalues as det(A - A m I) = 0,

(8.5)

where I is the unit matrix. The matrix A can be decomposed based on the sign of each eigenvalue Am A = A+ + A " ,

A± = R A ± R - 1 .

(8.6)

Here A + ( A - ) consists of the positive (negative) parts of each Am only, i. e. A± = diagiX*), 8.1.1

A+ = max(\m,0),

Characteristic

A - = mm(A m ,0).

(8.7)

variables

Equation (8.1) can be solved with a use of the characteristic variables =R~1u(x,t).

(8.8)

v, t + Av,x = 0.

(8.9)

v(x,t) From Eq. (8.1) we obtain then

As A is diagonal the above matrix equation contains q decoupled scalar equations which are solved by vm(x,t)

= vm(x-\mt,0),

m = l,2,---,q.

(8.10)

From Eq. (8.8) we get u(x,t)=Rv(x,t)

(8.11)

which is equivalent to «

u(x,t) = J2 vm(x,t)rm m=l

q

= Yl vm{x - A m t,0)r m . m=l

(8.12)

Linear system of one-dimensional

equations

129

As a consequence of that u(x,t) depends only on the initial data vm(x — Xmt, 0) at the q points x - Xmt. The curves x — XQ + Xmt are called pcharacteristics. In the case of A is a constant matrix p-characteristics are straight lines. 8.1.2

Riemann

problem for the linear

equations

The characteristic variables are very useful for solving the Riemann problem uj u(x,t = 0) = | u r

for x < 0, for x > 0,

(8.13)

with

m=l

(8.14)

Ur = E < m=l

such that vp

vm(x,t = 0) = i t;^

for x < 0, for X > 0.

(8.15)

This implies for x-Xmt< 0, for a; - Xmt > 0.

(8.16)

If we design M(x, t) to be the maximum of m for which x — Xmt > 0, then M(x,t)

q

u(x,t)= J2 <*m+ E 7n=l

^rm.

(8.17)

m=M+l

This equation can alternatively be rewritten as follows: u(x,t)=ul+

J2 m

\ <x/t

{vm-vDvm=xxr-

Y,

(,<-vT)rm.

(8.18)

m

X >x/t

This solution consists of q waves, which are discontinuities propagating at the characteristic velocities, A m , m = 1,2, • • •, q, of the system. It is noteworthy that the solution of the Riemann problem is self-similar, i. e. u(x, t) = u ( a i , at) for a > 0.

130

8.1.3

Numerical

schemes for a system of one-dimensional

The wave propagation

hyperbolic

equations

method

We present the wave propagation method that was developed by LeVeque (1990, 1997a,b) for a linear hyperbolic system of equations of Eq. (8.1). Then, the difference Aiij, evaluated at the spatial point a;,, can be written as

AUi = va - Ui_! = Y, < r r = E w r> m=l

(8-19)

m=l

m

m

where WJ™ is the wave, X is its speed and a is the scalar coefficient which measures a strength of the wave m such that at = R _ 1 A u j . Let u° denotes the value at the interface between Uj_i and u;. Then, with the use of (8.19) we can write W

U? - 1H-! = E

U

< ~ U°i = E

">

AT"<0

W

^-

(8-2°)

AT">0

The flux at the interface can be expressed twofold: f(uj)

=

x w

AuJ = A u i _ ! + Y,

? ?

= A u « - i + A " A m , (8-21)

AJ"<0

f (u?)

=

Au° = Am - E

< C W r = A U i - A+Aui,

(8.22)

AfX)

where A± is defined by Eq. (8.6). Equation (8.1) can be discretized as < + 1 = U

""A^(fi+1~fi)'

(8-23)

where the flux U = f(u°) = A u ° .

(8.24)

Using Eqs. (8.21) and (8.22) for evaluation of fi+i and fj, respectively, we obtain fi = f(m) - A+Aui,

fi+1 = f( U i ) + A - A t i i + i .

(8.25)

Nonlinear system of one-dimensional

equations

131

Equation (8.23) can now be rewritten as u ? + 1 = u? - - ^ ( A - A u i + 1 + A+Aui) =11? + A?pwind.

(8.26)

This scheme is first-order accurate in space and is called the Godunov scheme (Godunov 1959). A second-order correction to Eq. (8.26) can be obtained by altering its right-hand side as (LeVeque 1997a,b) < + 1 = u? + A»pwind

- | £ ( F I + 1 - f,),

(8.27)

where the flux IATI ( l - ^ l A r i ) W r -

fi = | i m=l

^

(8-28)

'

Here Wj™ is a limited version of the wave WJ™, obtained upwindly by comparing Wm to WJ!^ if \f > 0 or to WJ+! if \f < 0. 8.2 8.2.1

Nonlinear system of one-dimensional equations Flux-difference

splitting

scheme

We consider the following set of nonlinear equations: u,t + f,x = 0,

(8.29)

where f is a nonlinear flux. An upwind scheme that is based on fluxdifference splitting decomposes the spatial difference of u and f into linear combinations of waves (see Eqs. (8.20) and (8.25)) Au

=

u?+1-u? = ; £ a r r r = £ m

Af = *s.i-*r = E w r r .

W

? \

(8.30)

m

(8.31)

m

where rm, am, and Xm are respectively the right eigenvector, wave strength, and the eigenvalue of the m-th wave component, evaluated at Xi. The symbol A indicates a difference between neighboring nodal points. The system of Eq. (8.29) can be rewritten in the quasilinear form u,, + A(u)u i X = 0,

(8.32)

132

Numerical schemes for a system of one-dimensional

hyperbolic

equations

where A(u) is the q x q Jacobian matrix which is defined as the derivative of f with respect to u, so that df = Adu.

(8.33)

This differential relation can be replaced by its finite-difference analog, namely, Af = AAu.

(8.34)

Roe (1981) showed how to construct a mean value A such that the above equation holds exactly for arbitrary pairs of state vectors. For the Roe scheme see Sec. 8.5.2. Fluctuation splitting requires that the matrix A be split into its negative and positive parts, i. e. A = A-+A+.

(8.35)

For the linear equations A * are denned by Eq. (8.6). Then, Af = A - A u + A+Au.

(8.36)

This expression can be compared with Eq. (8.25). The first-order formula for Eq. (8.29) becomes n+l _ „ n

u,

At

= < - ^(fS-i/2-C-i/2).

(8-37)

where f£_i/2 and f,^_1/2 denote fluxes at the left and right sides of the interfaces. We can find three equivalent formula for the interface flux: fi+1/2

=

5 + A " Au,

(8.38)

fi+1/2

=

fi+i-A+Au,

(8.39)

fi+i/2

=

i(fi + f i + i ) - | A | A u ,

(8.40)

where |A| = A + — A - . In practice Eq. (8.40) is the best choice because of its symmetry condition. The flux fj+i/2 is then

fi+i/2 = Uti + fi+i - £ lATIarrJ").

(8-41)

Nonlinear system of one-dimensional equations

133

As the spectral decomposition of the matrix |A| can sometime be cumbersome, Degond et al. (1999) developed the so-called polynomial upwind scheme in which |A| was replaced by Pfc(|A|), where Pk is a polynomial of degree k. 8.2.2

Euler

equations

If we introduce the vector u(a;, t) such that

i(x,t) =

QV

ui \ / «2 = us / \

g{x,t) Qv(x,t) E(x,t)

(

«a

„2

(8.42)

\ (8.43)

| ( 3 - 7 ) g + (7-l)«3

QV +p v(E + p)

5?(7»3 - V § )

V

J

system of hydrodynamic Eqs. (2.34)-(2.36) can be written in the form of Eq. (8.32). The Jacobian matrix A = f u is given as /

0 •|(3-7)|

1

0

(3-7)^

7-1

\ (8.44)

The eigenvalues of this matrix are: \1/ x

N

U

.2/

2

A (u) =

cs,

N

z

U

2

A (u) = —.

«1

V(u)

W2

=

r ± + Ci

(8.45)

Ui

«1

where c. =

is the sound speed. These eigenvalues are associated with the fact that information from any point in the flow propagates according to the equations dx

(8.46)

134

Numerical schemes for a system of one-dimensional hyperbolic equations

and ^

= f±c..

(8.47)

Equation (8.46) defines a trajectory To, along which the entropy s is constant and which follows a particle path. Variations in the entropy are convected according to the equation stt + vs,x = 0.

(8.48)

Equation (8.47) defines the Riemann invariants itt = «± / — ,

(8-49)

which are constant along the trajectories T± of Eq. (8.47). These trajectories follow forward and backward sound waves in the frame moving with the speed v. For the Euler equations, we have the following diagonalizable equation: R _ 1 A R = A,

(8.50)

with l \ v + cs H + vcs j

R

where:

Q

is the enthalpy density and E is the total energy density. The matrix A is / A x 0 A = 0 A

\ 0 0

2

0\ 0 =

/ v-cs

0 0 v

0\ 0 .

A3/

\

0 0 v + cs J

This equation proves that the Euler equations are hyperbolic. From this equation it follows that infinitesimal changes propagate along characteristics with speeds v - cs, v, v + cs. Note that at the points where cs — 0 all eigenvalues coincide and the system is not strictly hyperbolic there.

The shock tube problem

135

It is noteworthy that the right eigenvector

r 2 (u) = I

1 ^ "1 UX

\


(8.51)

which corresponds to A 2 (u), is linearly degenerate. Since -^,-,0)

(8.52)

we find that A 2 , u • r 2 = 0. So, with r 2 is not associated neither shocks nor rarefaction waves but contact discontinuities across which there is a jump in the mass density but the gas pressure and flow velocity are smooth. The other eigenvalues, r 1 and r 3 , might be either shocks or rarefaction waves, depending on \ii and ur. For the shock wave all of the state variables are discontinuous. For the rarefaction wave all of the state variables are continuous. While numerically solving the Euler equations it is important to discuss the shock tube problem which takes place at every interface of two neighbour cells.

8.3

The shock tube problem

The shock tube problem can be described as follows. Imagine, there is a thin tube filled with gas that is initially divided by a membrane into two different states. The gas has a higher density and pressure in one half of the tube than in the other half and the gas is motionless. At t = 0, the membrane is rapidly removed and the gas is allowed to flow, involving three distinct waves: a contact discontinuity (c) in the middle and a shock (s) or a rarefaction wave (r) at the left and the right sides, respectively. For the contact discontinuity the mass density is discontinuous but the pressure and velocity are continuous. The rarefaction wave propagates in the opposite direction to the shock, with the mass density decreasing as the wave passes through. The fluid is accelerated abruptly across the shock wave and smoothly through the rarefaction wave (LeVeque 1997a). Four wave patters are possible: scs, scr, res, rcr. A nth pattern, which consists of a vacuum between two central contact discontinuities that are

136

Numerical schemes for a system of one-dimensional hyperbolic equations

surrounded by two rarefaction waves, is theoretically possible but it can never be realized in practice. A more complex shock tube problem results for real gases. For instance, for the van der Waals gas a mixed rarefaction wave can arise (Guardone, Vigevano 2002). In this wave the rarefaction fan is connected with a rarefaction shock. A classical compressive shock and a contact discontinuity propagate toward the low pressure region. A particular problem corresponding to Qi — pi — 3 and gr = pr — 1 is called the Sod problem (Sod 1978). As the pressure on the left, pi, is higher than the right one a rightwardly propagating shock wave results. The contact discontinuity which is easily seen on the mass density profiles propagates rightwardly and a rarefaction wave moves leftward.

8.4

Rankine-Hugoniot j u m p condition

A discontinuity in a solution of Eq. (8.29) propagates with the speed c which depends on the jump in the solution u(x, t) across the discontinuity. Suppose a discontinuity moves from left to right. At time t — t\ the discontinuity is at the spatial position x = x\ and at t = t2 at the point x = x2, where x\ < x2 and t\ < t2. Let the values of u be given as u/ on the left hand side of the discontinuity and as u r on the right hand side of it. The system of nonlinear conservation laws of Eq. (8.29) can be integrated to yield VLI{X2 -xi)

-ur{x2

- x i ) + f(u()(t 2 - * i ) - f ( u r ) ( t 2 -h).

(8.53)

In the limit x2 -> x\ and t2 -> t\ with c=

x2 - xi t2-h

we have c(u r - U() = f(u r ) - f(u,).

(8.54)

This relation between the shock speed c and the states U( and u r is called the Rankine-Hugoniot jump condition. For the case of scalar equations with ui and ur we get

c=/M-/M 7/._ — 11.1

(855)

The Riemann

problem for the Euler

equations

137

This equation suggests that any jump is allowed, provided the speed c is related via the above formula. For the case of a system of equations, we define vectors [u] = u r — u/ and [f] = f (u r ) — f (uj) while c is a scalar. Now, only certain jumps u r — u; are allowed, namely those for which vectors [u] and [f] are parallel one to each other. For a system of linear equations with f (u) = Au, Eq. (8.54) leads to c[u] = A[u].

(8.56)

As a consequence of that [u] and c are respectively an eigenvector and the associated eigenvalue of the matrix A.

8.5

The Riemann problem for the Euler equations

In the Riemann problem, an imaginary membrane, which separates two cells at different states is ruptured, and shock, contact discontinuity, and rarefaction waves are emitted when these two states interact. In other words the Riemann solver is based on the idea that two adjacent arbitrary states will evolve into a set of left- and right-going shocks and rarefactions. With certain assumptions on the flux function f(u), it is always possible, in principle, to solve the Riemann problem if the states u/ and u r are sufficiently close to each other. The solution consists of waves traveling with finite velocities. These waves may either be discontinuous shock waves or smooth rarefaction waves. The procedure for constructing the solution of a Riemann problem is called a Riemann solver. The most popular Riemann solver is due to Roe (1981). For the Euler equations, we are looking for states uj? and u* and speeds ci < C2 < C3 such that the Rankine-Hugoniot condition of Eq. (8.54) is satisfied «-u()

=

f«)-f(u,),

(8.57)

c2(u;-u?)

=

f«)-f(u?),

(8.58)

caK-O

= fK)-f«).

(8.59)

C l

It can be shown that ci — IJ(UJ*) = v(u*), where v is the flow speed (LeVeque 1990). This means that the second discontinuity propagates with the flow speed A2 = v. Moreover, as p(uf) = p(u*) the pressure is continuous

138

Numerical schemes for a system of one-dimensional

hyperbolic

equations

across this discontinuity which is known as contact discontinuity. We remind that across a contact discontinuity mass density, energy, and entropy are discontinuous, but pressure and velocity are continuous. Across shocks all dependent variables change discontinuosly. Shocks propagate with a velocity which is uniquely determined through the Rankine-Hugoniot relations of Eq. (8.54). The first discontinuity would be a shock if the entropy condition is fulfilled (v - cs)(ui) > ci > (v -

(8.60)

cs){ui).

Otherwise, this jump would be a rarefaction wave. Analogously, the third discontinuity is a shock if (y + cs)(u*T) > c 3 > (v + c s )(u r ).

(8.61)

A solution of the Riemann problem is shown schematically in the x- and ^-coordinates in Fig. 8.1.

tt

contact

discontinuity

v

X

Fig. 8.1 Wave structure of the Riemann problem for the one-dimensional Euler equations. The Riemann problem can be solved exactly with the use of RankineHugoniot relations across shocks and the isentropic characteristic equations across rarefaction waves (Gottlieb, Groth 1988). Such procedure leads to a nonlinear algebraic equation (for a flow variable) which can be solved by a Newton iterative method. As a consequence of that it is clearly a computationally expensive procedure and therefore different methods are required. These methods are described in the following part of the monograph.

The Riemann problem for the Euler equations

8.5.1

The HLL Riemann

139

solver

A simple approximate Riemann solver was developed by Harten, Lax, and van Leer (1983). In this solver (hearafter called HLL) the solution is approximated by two waves which propagate with their speeds c~ and c + such that they correspond to the minimum and maximum characteristic speeds of the system (Einfeldt 1988). The wave strengths in the HLL solver are: W 1 = u , - uj, W

2

(8.62)

= u r - u*,

(8.63)

with the middle state u* that is chosen to preserve conservation (c+ - c _ ) u , = c+u r - c~ui - (f(u r ) - f(uj)).

(8.64)

Hence the intermediate state is obtained as u, =

c+

* c _ [c+ur - c-u, - (f(u r ) - f(u,))].

(8.65)

As the Euler equations evolve three distinctive waves, with its speeds given by Eq. (8.45), obviously the HLL Riemann solver suffers a drawback and a more appropriate solver is required. 8.5.2

The Roe approximate

Riemann

solver

We consider a one-dimensional Riemann problem at the cell edge for the system, given by Eq. (8.29). At the left side of the edge, there is the state uj, on the right side u r . The solution of the Riemann problem for a nonlinear hyperbolic system like the Euler equations in general needs iterative methods which are not very efficient. For small jumps at the interfaces, it is sufficient to use an approximate Riemann solver which is based on a local replacement of the nonlinear equations by a linear hyperbolic system. To appropriately solve the Riemann problem, we can use the following linearized equation: u t + A(u)u,„ = u, ( + A u „ = 0.

(8.66)

This idea is used in the Roe scheme (Roe 1981) the keystone of which is the introduction of an average Jacobian A, which approximates the Jacobian A = f u , associated with the one-dimensional hyperbolic system of conservation laws of Eq. (8.29). The average Jacobian (called also the Roe

140

Numerical schemes for a system of one-dimensional hyperbolic equations

matrix) is such that for any given left and right pair of states (u;,u r ) the so-called Property U is satisfied: (i) A is a linear mapping from the vector space u to the vector space f; (ii) A ( u j , u r ) —> f u as u/ and u r —> u; (iii) A ( u j , u r ) has real eigenvalues and a complete set of linearly independent eigenvectors; (iv) A ( u r - u/) = fr —ftfor any u; and u r . In the original Roe scheme, the average state u used to linearize the problem is not (u/ + u r ) / 2 . Instead it is taken such as the property (iv) is satisfied. In the case of Euler or MHD equations, average mass density is given by Q =
(8.67)

and the rest of flow variables (v, E), which are denoted here by <j>, are averaged as follows (e. g., Asian 1996):

^ V a f t +Vfr^

(868)

y/0l + y/(fr

Once all the averaged variables are obtained, the linearized Riemann problem, expressed by Eq. (8.66) is considered at each interface. The exact solution of this approximate problem can be expressed in terms of right eigenvector r m of A as g

Au = u r - u, = Y,

Q,

"r m -

(8-69)

m=l

Here, q corresponds to the number of eigenvectors. The coefficients am may be determined by multiplying the above equation by each left eigenvector V. Noting that

the respective am is defined by

In the above equations the eigenvalues of A are the wavespeeds, the right eigenvectors define the paths taken in phase-space by simple waves. The left eigenvectors define the characteristic equations.

The Riemann problem for the Euler equations

141

According to property (iv), with a use of Eq. (8.69) we get the vector flux increment expressed as a product of Au and the corresponding eigenvalues Am, viz. Q

am\mTm.

Af = fr - f, = ^

(8.71)

771=1

8.5.3

A relaxation

Riemann

solver

In the relaxation scheme of Jin and Xin (1995) Eq. (8.29) is replaced by a system of coupled equations u,t + v, s

=

0,

(8.72)

v, t + B 2 u,x

=

I(f(u)-v), r

(8.73)

where u, v 6 Rq and B 2 6 Rqxq is a positive definite matrix. In the original scheme this matrix was chosen to be a diagonal matrix with positive diagonal elements (Jin, Xin 1995). The relaxation time is denoted by r > 0. As T - • 0 from Eq. (8.73) we get v -> f (u)

(8.74)

if a subchajracteristic condition is satisfied |A| < bmax, where bmax = maxm{bm} values of B such that

(8.75)

is the spectral radius of B with positive eigen-

6m>0,

m = l,2,---,q.

(8.76)

In Eq. (8.75) A is an eigenvalue of the Jacobian matrix f u . Equations (8.72) and (8.73) can be advanced with a use of a fractional step method of Sec. 6.8.1. First the following equations are advanced over time step At: u t + v, x = 0, 2

v, t + B u,« = 0.

(8.77) (8.78)

142

Numerical

schemes for a system of one-dimensional

hyperbolic

equations

This step leads to u* and v* which are updated to u n + 1 and v™+1 by solving the equations ut

=

0,

(8.79)

vt

=

-(f(u)-v).

(8.80)

T

Prom Eq. (8.79) we have un+1=u*.

(8.81)

Equation (8.80) can be treated implicitly to obtain v n+l =

f ( u n+l)

+ e-At/r

j y . _ f ( u „+l)] _

(882)

In the limit of T - • 0 this expression simplifies to v"+1 = f(un+1).

(8.83)

In summary, the relaxation scheme consists of solving Eq. (8.77) and using Eq. (8.83). An approximate Riemann solver is defined then as follows. Given values u/ and u r we compute vj = f(uj) and v r — f(u r ) and then solve the Riemann problem for Eq. (8.72) with the data (LeVeque, Pelanti 2001)

{«;>. { « . 8.5.4

Extension of state

of the Roe scheme for a general

<8M) equation

The Roe scheme was originally devised for a perfect gas. This scheme is valid for fluid of a constant ratio of specific heats, 7. One way to extend this scheme for a variable 7 is to adopt a mean value of 7. The other options are based on application of some sort of averaging (Glaister 1988, Toumi 1992, Saurel, Larini, Loraud 1994, Guardone, Vigevano 2002). Here, we describe briefly the method which was developed by Hanawa, Nakajima, and Nobuta (1999). It is useful to introduce the specific total enthalpy H and the specific enthalpy h as 2 V

H = E + ?- = h+ — Q

2

(8.85)

The Riemann problem for the Euler equations

143

and assume that thermodynamic variables depend on the mass density Q and the specific internal energy e. The Jacobian matrix possesses the right eigenvectors

(8.86)

\ H — vc J and the corresponding eigenvalues A1 = v + c,

A2 = v,

A3 = v - c,

(8.87)

where c is the averaged sound speed such as c2 = (

7

-l)^-ii)

2

)

(8.88)

and e =

Qi+lEi+1

- QiEi - ^{pi+1 - Pi) — . Qi+i ~ Qi~ &(Pi+i -Pi)

(8.89)

The other averaged quantities are Q = v

=

H

=

s/QiQi+i, y/QiVi + y/Qi+iVi+i v -*— —, \/Qi + y/Qi+l

(8-90) (8.91)

\fcHi + V^Hi±L

(8 .92)

yfQ~i + y/Qi+1

with 1

7-1

_

y/Qjhj +

^ | ,

+

y/Qj+lhj+i

^ |

+

(8.93) 1

The derivative | 2 is evaluated at the constant entropy. The eigenvectors r 1 and r 3 correspond to the sound waves and the eigenvector r 2 is associated with the entropy wave. The amplitude of each

144

Numerical schemes for a system of one-dimensional hyperbolic equations

wave is given as a1

=

vi+1-vi

+

Pi+i Pi-Pi Pi+1

__ , gc QC

2

_

=

Qi+i-Qi

,

(8.95)

Q3

=

-v^x+^ + ^ r ^ .

(8.96)

a

Pi+1 ~ Pi

(8.94)

=5

This scheme constists a natural generalization of the Roe scheme (Roe 1981). In the case of the perfect gas law p — (7 - l)ge and the scheme reduces to the Roe scheme.

8.6

Deficiences of Godunov-type schemes

Godunov-type schemes are very robust and give reliable results for a wide range of problems without needing to be retuned. However, even these modern schemes are far from being perfect. There are few instances in which a particular scheme produces inappropriate results (Donat, Marquina 1996, Einfeld et al. 1991). For instance, most Godunov-type schemes lead to the generation of a long wavelength noise, downstream nearly stationary shock. This noise is not effectively damped by the dissipation of the scheme (Donat, Marquina 1996). In few cases Roe solvers exhibit nonlinear instability, producing unphysical local features which are called carbuncles (Quirk 1994b). These features become more pronounced for finer grid. In multi-dimensions a problem occurs if a wave is far from aligned with the grid. Then, a grid-oblique wave may be represented by grid-aligned waves, enhancing numerical dissipation and leading to a loss of resolution. Additionally, the Roe method can admit spurious solutions that are triggered by an incorrect treatment of shear waves. A weighted average flux formulation was used by Quirk (1991) to devise a shear fix for Roe's method. 8.6.1

Entropy

fix

Another deficiency of a Godunov-type scheme is that while computing rarefaction waves, the scheme can produce nonphysical expansion shocks in the computed flow. In this case the true Riemann solution contains a transonic wave with characteristic speeds that increase from negative to positive

Deficiences of Godunov-type schemes

145

values through the rarefaction fan. Then, the eigenvalue of the average Jacobian A is such that Xi < 0 to the left of the wave while Ar > 0 to the right of the wave. It leads to information travelled partly to the left and partly to the right, affecting cell averages on both sides. The Roe solver approximates every wave by a single discontinuity that propagates at a speed given by an eigenvalue c of A. In the transonic rarefaction case this speed is approximately zero and the proper spreading does not occur. This can lead to numerical approximations with entropy violating discontinuities. Several ways to fix the problem of prevention Roe method from admitting expansion shocks exist in the literature. For instance, Yee, Warming, and Harten (1985) replaced values of the numerical viscosity \i smaller than some tolerance e with higher values /J,' such that fn

for \(i\ >e,

" " { t ^ + i) * » w < -

(8 97)

'

For typical simulations e is set to 0.2. This modification is only applied to rarefaction waves. Although the dependence on e is small and this entropy works well it suffers from a drawback that a tunable parameter e was introduced into the scheme and there is little physical justification for its use. To prevent the expansion shocks, an intermediate state (that simulates the diffusion) between the left and right states is introduced (Harten, Hyman 1983). More precisely, the single wave ai (coming from the Roe solver) is replaced by a pair of waves ajr and a r r , propagating at speeds A/ < 0 < A r . These speeds are chosen to approximate the characteristic speeds at each edge of the rarefaction fan. As the total wave strength should remain the same, we require an + ar = a.

(8.98)

To maintain conservation we also need aiXi + arXr = ac.

(8.99)

From these equations we obtain aXr — c Ar — A; _ c — aXi Xr — A;

(8.100) (8.101)

146

Numerical schemes for a system of one-dimensional

hyperbolic

equations

The above procedure is called an entropy fix. This procedure is necessary to obtain physically relevant numerical approximations of the exact solution. In Roe method (Roe 1981) one only needs an entropy fix at sonic points (Roe 1982). There was some debate if this is the same for the magnetohydrodynamic waves. However, the simplest approach is to apply the entropy fix for the magnetosonic (fast and slow) waves only. The Alfven and entropy waves are supposed not to need entropy fixes as they are linearly degenerate. A very popular entropy fix method was developed by LeVeque (1990). The idea used in the entropy fix is to replace the single jump u r — uj, propagating at speed A by two jumps propagating at speeds Aj and Ar, with a new state u* in between (Harten, Hyman 1983). The flux difference can be expressed as follows: f(uj) - f(u r ) = f(uj) - f(u.) + f(u.) - f(u r ).

(8.102)

Using the property (iv) of the Roe solver this formula can be rewritten as A(uj - u r ) = A/(u* — u«) + Ar(u» - u r ) .

(8.103)

Hence, _ (A - Aj)uj -I- (Ar - A)u r Ar — A;

(8.104)

and consequently u* - uj =

Ar — A . . - ( u r - uj) = AT

Ur -

U* =

Aj

A - A; Ar

-(Ur

Aj

Ar — A 3 ,•J .• -a r , Ar

-

Uj)

=

A — A/ Ar

(8.105)

Ai

Ai

• •

-a3T°.

(8.106)

The flux can now be written twofolds (LeVeque 1990) f(uj,u r ) = f(uj) + J2 A~amrm

+

Haj*j

f(uj,u r ) = f(ur) - J^ X+amrm

- A^r^,

(8.107)

or

m^tj

(8.108)

Deficiences of Godunov-type

schemes

147

where we introduced the following notation:

^A'T^T-

M^^^Y' Ar

Al

( 8 - 109 )

Ar — A{

The eigenvalue X± is denned by Eq. (8.7). Equations (8.107) and (8.108) are used instead of Eqs. (8.21) and (8.22). Another entropy fix that was based on estimates for the spreading rate of each wave within the approximate Riemann solution was devised by Roe (1985). Moreover, Osher (1984) found a general condition for a scheme to be entropy satisfying when applied to scalar equations. He designed such schemes which were called E-schemes. The above presented entropy fixes may fail in some circumstances. For instance, it has been found that the above entropy fix failed in the presence of a negative/positive transonic rarefaction of the van der Waals gas (Guardone, Vigevano 2002). In this case the standard entropy fix of Harten and Hyman (1983) overcome this problem (Guardone, Vigevano 2002).

Chapter 9

A hyperbolic system of two-dimensional equations

We consider a two-dimensional wave problem which is described by a conservative system of equations: u i t + V - F = 0,

(9.1)

where u stands for a vector of conserved variables (u±,U2,- • • >U«)T a n d F is the flux which is a q x 3-vector. Integrating Eq. (9.1) with respect to time over time-step At, and with respect to space over a cell surface Ax Ay, we find the following discretization:

<+1=<-A^£F(fl-)-lm'

(9 2)

-

where Gauss' theorem has been applied to replace integration over surface by integration over the cell edges. The vector l m is normal to the cell edge m and has the length of this edge. In this case the cell of four edges was chosen. The quantity u ^ stands for the average of cell ij, taken at time-step nAt, and u ^ denotes u n which is evaluated at the cell edge m. It is noteworthy that discretization (9.2) mimics the integral form of Eq. (9.1). This property is of vital importance as the discretization should be able to capture shocks. Differential form (9.1) does not make sense for discontinuous profiles. However, the corresponding integral form is still valid. The surface integral, J F • dl, in Eq. (9.2) has been approximated by taking at each cell edge an average flux F = F ( u ^ ) , with u ^ a suitably chosen average for the particular cell edge m. The most accurate choice for 149

150

A hyperbolic system of two-dimensional

equations

uj^ would be to take the average of the left u " and right u™ values, i. e.

However, with this average Eq. (9.2) attains in some way the equivalent form of the central discretization (6.14) for the advection equation, which is second-order accurate but unstable scheme. This problem can be removed by the application of an operator splitting method which is described in the forthcoming section.

9.1

Operator splitting schemes

A common approach when solving multi-dimensional hyperbolic equations is to apply an operator splitting method (e. g., Murawski, Goossens 1994a,b) The idea of the operator splitting method is to iterate sequentially one dimensional equations; in each time-step, multi-dimensional derivatives are split into a set of one-dimensional derivatives, with variations in other directions ignored temporarily. Then, each row and column in the grid is treated as if it were a one-dimensional problem. Updating the flow quantities along each row is done using the one-dimensional solver. The popularity of these methods is a consequence of the fact that the numerical schemes lead to surprisingly good results (e. g., Stone, Norman 1992) and that the strategy is very simple as any multi-dimensional scheme consists of a system of the one-dimensional problems. We explain this strategy for the two-dimensional system of equations u t + Au

x

+ Buy = 0

(9.4)

which can be split into one-dimensional subequations: u )( + Au ]X

=

0,

x-sweep,

(9.5)

u)t + B u „

=

0,

y—sweep.

(9.6)

In the x-sweep we would solve Eq. (9.5) along y — const, updating u to u*. In the y-sweep we then use u* for solving Eq. (9.6) along x = const. Such procedure will generally introduce a splitting error unless A B = B A . We explain this in the following way. Define the exact solution operator SA* as the operator which advances the exact solution u(a;, y, t) of Eq. (9.4) by a time-step At. Analogously S A t and S A ( are the operators

Operator unsplit methods

151

which advance exact solutions of Eq. (9.5) and (9.6), respectively. With a use of the Taylor expansion these operators can be written as S^~e-AtA&,

S^~e-AtB&,

SAt~e-A'(A£+B&>,

(9.7)

where we introduced the notation e

"

=

1 + a

^

+

2 ! ^

+

3 !

( a

^

+

- -

(9 8)

-

Strang (1968) showed that S i i S ^ S i , = S A ( + 0((Ai) 3 ). 2

(9.9)

2

So, the above combination of one-dimensional solution operators approximates the full evolution operator within second-order accuracy. The splitting error does not often exceed other numerical errors, and the dimensional splitting can be a very effective approach (e. g., Murawski, Goossens 1994b). However, the operator splitting methods have several disadvantages. For example, discontinuities traveling obliquelly to the grid are smeared more than those traveling in the coordinate directions. The implementation of boundary conditions may also be complicated using this method.

9.2

Operator unsplit methods

In unsplit methods, information is propagated in a genuinely multi-dimensional way. One-dimensional Riemann problems can be solved at the interfaces. Limiter functions are applied to suppress numerically induced oscillations which are usually generated by higher-order derivative terms. The left-going and right-going waves are split into parts propagating in the transverse direction by solving Riemann problems in coordinate directions tangential to the interfaces. These cross-derivative terms are necessary for obtaining both stable and second-order schemes (LeVeque 1997b). A class of conservative finite difference schemes for hyperbolic conservation laws in multi-dimensional spaces has been developed by Colella (1990). These schemes do not make use of operator splitting and instead the multidimensional wave properties of the solution are used to calculate fluxes. In these schemes some of the second-order terms are limited to suppress oscillations. Although the same Riemann problems appear in these schemes

152

A hyperbolic system of two-dimensional equations

as in the operator split methods, these schemes are somewhat more expensive, requiring twice as many solutions to the Riemann problems as the corresponding operator split algorithm.

9.3

Grid generation

One of the basic problems in numerical discretisations is grid generation (e. g., Jordan, Spaulding 1993). The purpose of a grid is to provide a skeleton upon which some discretized version of the equations may be solved numerically. The ideal grid would be one that is simple to implement, cheap to generate, places no unreasonable constraints on the method of solution, is able to match the local grid spacing to the local spatial length scale, and is able to represent complex flows.

9.3.1

Structured

and unstructured

grids

At present, there are basically two approaches for numerical simulations. One is based on the structured grid and the other is unstructured grid (Barth 1990, Mavriplis 1996, Pirzadeh 1996). A structured grid is one which can be mapped from physical space to a computational space in which it appears as a rectangle, in two dimensions. Similarly, an unstructured grid is one in which there is no mapping from the physical space to a simple computational space. A simple example of a structured grid is a cartesian mesh in which all points are regularly distributed at equal distances from one another throughout the flow field. Among the unstructured grid alternatives, there is proliferation of schemes using triangles in two dimensions and prisms, hexahedra or tetrahedra in three dimensions. In contrast to the structured grid approach, these elements are not ordered in a regular fashion but they fit the boundaries of the simulation domain. For instance, for the numerical domain which is subdivided into triangular elements, the mass conservation equation can be written as / Jv

PitdV

+ f (f + g) • ndl = s, Jdv

(9.10)

where n stands for the outward unit normal vector, f and g are respectively the fluxes in the x- and y-directions, and s is a source term. The semi-

Grid generation

153

discrete counterpart of this equation is 3

VPtt + Y,Fi^h=0,

(9.11)

i=l

where V denotes the volume of the cell and Fi is a numerical approximation of the normal flux crossing a given interface i with side length AZj. Using Roe scheme (Roe 1981) Fi can be expressed as Fi(pi,Pr)

= \[F{Pi) + F(Pr) - \A\(Pl -

Pr)},

(9.12)

where A represents Roe linearized Jacobian matrix and the subscripts / and r indicate the left and right states which share the face i. Development of unstructured grid methods has been accelerated from mid 80's. But even with these rapid improvements, the unstructured grid algorithms cannot attain superiority over the structured grid methods yet. This is due to the fact that the unstructured methods are inefficient in comparison to structured methods. As a consequence of the indirect addressing performed when computing on random data-sets, the performance on vector computers is usually lowered. In addition, many three-dimensional unstructured algorithms require excessive memory overheads which severely limit the size of the meshes. These problems of the unstructured grid methods become worse for high-Reynolds number viscous flow computations. Moreover, there is a difficulty to implement an implicit time integration. As a consequence of that, most of those applications are limited to inviscid flow computations at present. Both structured and unstructured grid approaches have their advantages and disadvantages. Structured grid methods have advantages in the accuracy of a solution and the computational efficiency especially for high Reynolds number viscous flows. The disadvantage of the methods is that they need an extremely large number of grid points to resolve the viscous boundary layers on curved surfaces. On the other hand, the unstructured grid methods are attractive for complicated geometries. This trend is mainly driven by the fact that the unstructured grids provide for the implementation of adaptive grid-flow coupling procedures. However, a disadvantage of unstructured grid method is a difficulty for an efficient parallel computation. As a consequence of that semi-unstructured grid methods were developed (e. g., Nakahashi 1995). These methods use a combination of structured grid and unstructured grid.

154

A hyperbolic system of two-dimensional

equations

The generation of an unstructured mesh for a complex geometry is usually accomplished in two phases. In the first, the boundaries of the domain are discretized to form the surface mesh. In the second phase, the volume mesh is generated filling the domain with unstructured grid. Some of these methods use portions of a structured or semi-structured mesh in the viscous regions which are then matched up with an unstructured mesh in the inviscid regions. 9.3.2

Other grid generation

methods

Two standard methods are commonly used for generating body-fitted structured grids: elliptic and algebraic grid generation. The former generation is based on solving Laplace's equation with smoothing the boundary data over the numerical domain (Thompson et al. 1982). The grid lines represent streamlines of potential flow. Algebraic grid generation is based on the idea of a smooth interpolation between points on boundary curves. Algebraic methods are faster than elliptic ones, but they are not so easily automated. Among sophisticated grid generation schemes we can also distinguish the Advancing- .Front method (Peraire et al. 1987). First, a list of frontal faces is created between boundary nodes. The smallest face typically becomes the start of the front. An ideal third node is created from this face and put in a new list of nodes. Then, all other nodes in the triangulation are sorted by their distance from the new node and added to the list. The first node on the list which creates a triangle without crossing existing faces is used. The front is then updated, and the process repeated until completion. Such created grid is usually smoothed with a Laplacian filter, resulting in a grid with a high degree of regularity. The other methods which are worth mentioning are Veronoi and Delaunay grids. The triangulation of the boundary nodes is taken as the initial grid. Cells with a high skewness are then refined by the insertion of a new node at the circumcenter of that triangle, followed by retriangulation. This procedure begins by dividing the domain into Voronoi regions (Voronoi 1908). The Voronoi region for a given node consists of the part of the plane which is closer to that node than any other (Fig. 9.1). A unique triangulation results when nodes whose Voronoi regions share a common boundary are connected. This procedure is called the Delaunay triangulation (Delaunay 1934). The Voronoi cells are unique and their boundaries consist of the perpendicular bisectors between points. The Delaunay triangulation is

Grid generation

155

unique unless four neighbors lie on a circle or five on a sphere. To make the Delaunay mesh useful an interpolation method is required to estimate the solution between the nodes. A simple linear interpolation leads to low-order schemes. Braun and Sambridge (1995) used natural-neighbor interpolation in which u(r) = ^Ni(r)ui,

(9.13)

i

where the weights iVi(r) are the natural-neighbor coordinates of r with respect to its natural-neighbor nodes such that Ni{vj) = Sij, and iij is the corresponding value of u at the node i. Brio, Zakharian, and Webb (2001) constructed a multi-state Riemann solver for the Euler equations which are discretized with the use of triangular grids or any other finite-volume tessellations of the plane.

Fig. 9.1 Voronoi cells (left panel) and Delaunay triangulation (right panel). The neighbors of node A are numbered 1-8. A disadvantage of the Delaunay triangulation is associated with its inability to guarantee boundary integrity unless the domain is convex. Other problems related to the Delaunay triangulation are listed out by Mavriplis (1996). Among them it is noteworthy that the process of regridding using Delaunay triangulation is computationally expensive. Moreover, the addition or the deletion of a single node can change the triangulation dramatically, perturbing invariably flow representation between grid adaptations. Finally, the storage overheads is considerably high. Both grid-point redistribution and grid embedding have been employed on structured grids which are especially well suited to grid-point redistribution. The solution of the given equation is computed on a grid which consists of a fixed number of points. These points can be redistributed in

156

A hyperbolic system of two-dimensional equations

the vicinity of flow features. The desired result is the best solution for a fixed cost. A numerical grid can be generated by Lagrangian or Eulerian methods. In the method of Lagrange'a the grid points can move together with fluid (e. g., Oran, Boris 1988). This method is useful as it eliminates convective terms from the equations. As a consequence of that it is easier to satisfy the positive mass requirements and fulfill the conservation laws. A serious problem in this method is created by complex flows which quickly distort the numerical grid. Accuracy of a computation is low there and the grid deformation occurs usually before a useful result is obtained. Another problem is associated with existence of the different convective terms in the mass and energy conservation equations. A grid can be Lagrangian for energy but not for mass density and vise verse. As a consequence of that it is impossible to eliminate all convective terms with a use of the same numerical grid. In Euler methods numerical grid does not move but it is static. Since the liquid flows through such stationary grid, the cut-off errors arise. These errors grow in time, leading to a poorer accuracy of the solution. For compressible flows such errors are the most important in a neighborhood of large gradients and complex flows. These errors can be minimized by implementation of inhomogeneous grid which is finer in the neighborhood of large gradient regions.

9.4

Adaptive mesh refinement method

If a numerical solution of a flow-field containing complex flows such as occur at high-gradient regions has to be determined, an appropriate resolution of these phenomena is of central importance for the overall quality of the solution. The positions of these high-gradient regions are usually unknown. At the same time it is not possible to refine the whole discretization domain because of limited computer power and required cost efficiency. A possible solution is to refine the grid locally where the high-gradient regions are detected (Berger, Colella 1989, Bell et al. 1994, Martin, Colella 2000). The adaptive grid methodology makes it possible to achieve very high resolution in the most interesting regions. The algorithm is particularly well suited to unsteady flows. The adaptive mesh refinement method (AMR) employs a hierarchical

Adaptive mesh refinement

method

157

grid structure which changes dynamically and which is composed of grids of varying resolution. The grid which covers the entire computational domain is called the level 0 grid. There are also few additional levels of grid, each finer than the rest. These finer grids do not cover the whole domain but only those regions where more resolution is determined to be needed. The mesh is refined locally based on an estimate of the solution error (De Zeeuw, Powell 1993). By concentrating mesh points where they are most needed, high-quality solutions can be obtained at reasonable computational cost. By that way, careful attention is paid to resolving efficiently the disparate length scales. Various refinement criteria may be employed. For example, the local velocity divergence can be used to detect compressive phenomena, whereas the velocity curl can be used to detect shear (De Zeeuw, Powell 1993). That is, for a cell with a characteristic size Aa;, the compressibility dc and shear d3 detectors are dc = | V - v | ( A z ) 3 / 2 ,

ds = |V x v|(Ax) 3 / 2 .

(9.14)

Cells are refined or coarsened if dc and d8 is above or below a specified threshold. Complex geometries can be treated with the use of the cut-cell approach (Quirk 1994a). In this case, a cartesian grid is superimposed on the physical domain. Grid points which fall outside of the flow field are discarded, resulting in a series of intersected cells along the boundary delimiting the flow field. While the discretization in the interior of the flow field is unchanged, the discretization at the boundaries must be altered to account for the cut-cells. For complex flows, adaptive meshing can easily be implemented by simple cell-subdivision. It is noteworthy here that the remeshing technique is relatively easy for triangular grids (Trepanie et al. 1993). The refinement can be performed through triangle subdivision, where a triangle is branched into two triangles by cutting it on its longest side. The coarsening can be obtained through node removal. The node is flagged for removal if all its neighboring triangles are to be coarsened. This node removal leaves an open polygon, which is then remeshed (Trepanie et al. 1993). 9.4.1

AMR

codes

In this subsection we describe briefly several codes which were recently developed with a use of AMR algorithms. An adaptive Cartesian mesh algorithm was developed by De Zeeuw and

158

A hyperbolic system of two-dimensional equations

Powell (1993). This algorithm was successfully applied to obtain steadystate solutions of the MHD equations for a solar wind interaction with comets (Gombosi et al. 1994, 1996). It is based on a MUSCL-type numerical technique which creates an initial uniform mesh and cuts the body out of that mesh to resolve high-gradient regions of the flow (De Zeeuw, Powell 1993). For a similar code see also Fischer and Hirschel (1993). An unsplit second-order Godunov algorithm was applied in a conservative adaptive mesh refinement scheme that selectively refines regions of the computational grid to achieve a desired level of accuracy (Pember et al. 1995). Examples showing the results of the combined Cartesian grid integration and adaptive mesh refinement algorithm for both two- and threedimensional flows suggest that the scheme may be globally second-order accurate and first-order accurate at the boundary in smooth flows. A Cartesian cell-based scheme for adaptively refined solutions of the Navier-Stokes equations in two-dimensions was presented by Coirier and Powell (1996). Grids, in this scheme, about geometrically complicated bodies are generated automatically by the recursive subdivision of a single cell that encompasses the entire computational domain. Where the resulting cells intersect bodies, polygonal cut cells are created.

9.5

Implicit hydrodynamic schemes

Historically, explicit schemes were developed earlier, as a consequence of their simplicity over implicit methods (Collins et al. 1995, Dai, Woodward 1996). However, the major limitations of explicit methods is their stability constraint which imposes an upper limit on the allowed time-step (see Sec. 6.5). Therefore, implicit methods were developed to overcome this limitation and have proved to be more efficient than the explicit methods. For many problems, the important physics occurs on time scales that are much longer than the characteristic transit time. In this case, an explicit scheme would limit the time-step to a much smaller value than is needed to accurately resolve the transient behavior. An implicit scheme removes the numerically imposed time-step constraint, allowing much larger time-steps (e. g., Korevaar, van Leer 1988). While using an implicit code, it is expected that as a result of acting numerical errors shocks are smeared out somewhat in a transient solution of implicit schemes. Therefore, if one's goal is to accurately track moving

Implicit hydrodynamic

schemes

159

shock fronts, explicit codes would be a better choice than implicit schemes. However, efficiency of such schemes is not only related to the mesh layout, but also depends heavily on the time integration scheme. It is well known that in terms of the allowable time-step implicit schemes are less restrictive than explicit schemes. Implicit schemes are well known in hydrodynamics. For an extended review of various implicit schemes see Yee (1989). In particular, an implicit flux-corrected transport algorithm was developed by Steinle and Morrow (1989). Several techniques were developed to solve implicit equations. For example, Eqs. (6.80) and (6.81) can be generalized to a system of equations

u t = f (u)

(9.15)

(1 - 0Atf, u (u n ))(u n + 1 - u n ) = Atf(u n ),

(9.16)

which can be solved as

wher f u is the Jacobian. This scheme produces a large sparse system of linear equations which need to be solved at each time-step in order to update the solution. This implicit formulation of the time integration leads to an algebraic set of equations. In the limit of an infinitely large time-step, the first term on the left-hand side vanishes, and Newton's method is recovered. For finite time-steps, this scheme can be relaxed by a point Gauss-Seidel (G-S) technique (e. g., Fischer, Hirschel 1993). Typically three to six iterations per time-step may be enough for adequate convergence; it depends upon the goodness of the initial estimate. The idea is to use the old solution (perhaps with a linearly projected correction over the time-step) as the initial guess for the new solution to the matrix equation. Iterative relaxations are then used to smooth and reduce the resulting errors. Jacobi relaxation (both regular and accelerated) in the context of multigrid elliptic solvers (Mavriplis 1995) and also for finite-element hydrocodes was used by DeVore (1996). As long as the initial guess is decent, in those cases typically 3-6 sweeps were used. Since G-S method uses continuous updating, it converges faster than Jacobi technique. The G-S method works pretty well, but the convergence rate is not great. As a result of the relatively slow convergence rate, there are really only certain types of problems for which the implicit scheme is advantageous.

160

A hyperbolic system of two-dimensional equations

The time-steps must be many times the CFL limit to be efficient. This has some effect on shocks which will be smeared. Nevertheless, this approach works well for problems in which there is a large range of densities, for example, and one is most interested in what is happening in the high density regions. The large time-step will smear profiles out in the low density (high speed) regions which are sort of vacuum and therefore are less important to be well resolved. A special version of the G-S method is Lower-Upper Symmetric GaussSeidel (LU-SGS) algorithm which was first demonstrated by Yoon and Jameson (1987) in solving the Euler and Navier-Stokes equations. This algorithm was based upon an LU decomposition method proposed by Jameson and Turkel (1981). Coirier (1991) showed that the LU-SGS scheme for approximate Newton iteration is slightly more efficient on a work basis than the diagonalized Beam-Warming (1976) scheme and requires less user effort to achieve an acceptable convergence rate. The particular application code that this was installed in was a structured, multi-block solver. The intention was to eventually compute chemically reacting flows, where it is necessary to solve the species equations implicitly, since they are numerically stiff. But, even without the species equations, a factored method is well suited for a structured mesh flow solver. The solver this code (PARC) had in it before was the diagonalized Beam-Warming scheme. This implies that the LU-SGS can run with a higher Courant number, and also is better suited for reacting flows (Coirier 1991). In other contexts there are better iterative methods than Gauss-Seidel, e. g. preconditioned conjugate gradients (Tanaka 1997), Newton-KrylovSchwartz (NKS) or multigrid (e. g., Venkatakrishnan, Mavriplis 1995). In the context of implicit time-stepping, however, one has a very good initial guess from the previous step and it may be possible to just take a couple sweeps with any iterative method, or to use something like dimensional splitting or alternating direction implicit scheme (ADI) methods (LeVeque 1996). As far as multigrids are concerned, these methods both are good iteration strategies, and both have situations where they are well-suited, and situations where they are not so well-suited. However, multigrid seems to be the most elegant, and has the most potential. Multigrid methods are described briefly in Sec. 6.3.2. Regarding overtures on NKS method (e. g., Cai, Keyes, Venkatakrishnan 1995, Knoll, McHugh, Keyes 1996), it can never beat multigrid, when multigrid is 'working', but it is much more robust than multigrid (which it can always annex as a preconditioner),

Implicit hydrodynamic schemes

161

when multigrid is stumbling. A fully implicit method for the steady state solution of the Euler equations was implemented by Trepanier et al. (1993), using Roe approximate Riemann solver in a finite-volume framework. A particular linearization, tailored for Roe scheme, was developed and a new implicit treatment of the boundary conditions was designed. The global sparse-matrix system was tackled using a direct LU solver. A triangular grid adaptation procedure was implemented with an error sensor based on the gradient of the flow variables. The remeshing was performed by means of local enrichment and coarsening of the grid. The main difficulty with implicit methods relates to the memory requirements of these techniques. In fact many implicit schemes incur storage overheads equivalent to two to three times the Jacobian matrix, making them particularly undesirable for three-dimensional problems. Semiimplicit (Sec. 6.6.2) and multigrid (Sec. 6.3.2) methods offer an alternative to implicit methods, incurring low memory overheads.

9.5.1

Barely implicit

scheme for the Euler

equations

As a particular example of implicit schemes we present the barely implicit scheme (BIS) which is motivated by the need to calculate subsonic flows accurately in which, by definition, the speeds of the characteristic flow are lower than the sound speed. Such conditions are encountered, for instance, in low-speed fuel injection in engines or air-pollution simulations. For the latter ones see Sec. 11.3. The obvious way to overcome the CFL time-step of Eq. (6.75), imposed by such speeds, is to apply a fully implicit scheme (e. g., Beam, Warming 1978). However, implicit schemes are relatively expensive. Casulli and Greenspan (1984) showed that it is not necessary to treat all of the terms in the Euler equations implicitly to avoid the time-steps constraint imposed by the CFL condition. They proved that only the pressure gradient term in the momentum equation and the velocity in the term - V • [(E + p)v] in the energy equation have to be treated implicitly. This idea was undertaken by Patnaik et al. (1987) who developed a BIS scheme in which the mass continuity and momentum equations are discretized as follows: =

-V-(e"vn),

(9.17)

A hyperbolic system of two-dimensional

162

0*V* -

equations

OnVn

e

—-JL

=

-V-(envBv»)-Vpn,

(9.18)

where the superscripts n and * denote old and predicted values, respectively. This stage is called an explicit predictor. The implicit momentum and energy equations are: „n+lvn+l _

£

nnvn

—£

=

Kn+1 — En =

-V-(Qnvnvn)-V[vpn+1

+

(l-v)pn](9.19)

-V-{(£n+p")[z/v"+1+(l-z/)vn]},(9.20)

where 1/2 < v < 1 and

The implicit terms are centered in time for v — 1/2. For v < 1/2, the method is unstable for sufficiently large At. When sound waves are not important, v can be set equal to one. Subtracting Eq. (9.18) from Eq. (9.19) we obtain Q

-

^

e

= -Wu{pn+1

- pn) = -V6P,

(9.21)

where 5p is the implicit pressure correction. The new velocity can be obtained from this equation by letting gn+1 = g*. Then, v n + 1 = -A±-V5p

+ v*.

(9.22)

A correction for the internal energy e is given as en+1 = , S* + en. (7 - l)v

(9.23)

Substituting this equation and expression (9.22) into equation (9.20) we get (En+pn V g*v*2 - gnvn2 + 7 Sp W T = *N • -^~ V§P 2At (7 - T l)vAt " V Q* -vV • (En + p n ) v * - (1 - i/)V • (En + p n ) v n .

(9.24)

The energy can also be treated explicitly: -En

E* At

= - V • (En + pn) [w* + (1 - z/)v n ].

(9.25)

Few specific examples of hydrodynamic schemes

163

With the use of this definition Eq. (9.24) becomes the equation for Sp,

*P

(7 - l)vAt

_uAtv.(E^£.)VSp V Q*

=

E* - En

Q*\*2 - Qnwn2

At

2Kt

(9.26)

The right hand side of this equation is evaluated explicitly, using equations (9.25), (9.17), and (9.18). Having computed Sp, momentum and energy are obtained from Eqs. (9.21) and (9.20), respectively. These values and the mass density, which is obtained from Eqs. (9.21) and (9.22), are the starting conditions for the next time-step. As a consequence of that the cost of the method is comparable to the cost of an explicit method.

9.6

Few specific examples of hydrodynamic schemes

In this part of the monograph we present a few hydrodynamic schemes which were not described above and which are important in development of numerical methods for solving hyperbolic equations. However, this presentation is far from being complete. A numerical algorithm to solve the two-dimensional flow of a viscous, incompressible fluid in a branching channel was proposed by Bramley and Sloan (1987). This fluid was described by the Navier-Stokes equations which are solved in terms of stream function and vorticity using a finitedifference method. This algorithm was later extended by an application of a nonlinear multigrid algorithm (Lonsdale, Bramley, Sloan 1988). Numerical results show that this multigrid algorithm leads to a large reduction of computing time relative to the relaxation scheme that was used as a onegrid method. The wavelet theory (e. g., Kaiser 1994) was used by Charton and Perrier (1996) to construct the numerical scheme for the two-dimensional incompressible Navier-Stokes equations. Numerical simulations showed, by comparison to a Fourier method, that this scheme is efficient and accurate although time consuming. Explicit Chebyshev collocation method was applied by Wang, Nakamura, and Yasuhara (1993) to solve the initial-value problem for compressible Navier-Stokes equations. A numerical scheme for solving three-dimensional chemically reacting

164

A hyperbolic system of two-dimensional equations

flows which are described by the Navier-Stokes equations was developed by White, Korte, Gaffney (1993). The challenging problem in the computation of two-phase flows is the non-hyperbolic character of equations and non-conservative form of the system. This leads to an ill-posed initial-value problem which requires numerical damping to obtain stable results. Introducing a virtual mass term in the momentum equation is sufficient to make the model hyperbolic (Toumi, Kumbaro 1996). A three-dimensional adaptive mesh-refinement numerical scheme for multi-fluid hydrodynamic equations was constructed by Puckett and Saltzman (1992). A Godunov scheme for isothermal hydrodynamics was developed by Balsara (1994). An algorithm for low Mach number unsteady flows which are governed by the fully compressible Navier-Stokes equations was developed by Mary, Sagaout, and Deville (2000a,b). To remove the stiffness due to the large disparity between the flow and the acoustic wave speed at low Mach number, an approximate Newton method, based on artificial compressibility, was used. The stiffness is due to the fact that numerical stability considerations lead to small time steps for the acoustic waves, while the physics is mainly driven by the flow convection where the time scale is large.

Chapter 10

Numerical methods for the M H D equations

Plasma generally exhibits both collective (fluid-like) and individual (particlelike) behavior. In the magnetohydrodynamic model, the plasma is treated like a conducting fluid having macroscopic parameters that accurately describe its particle-like interactions (e. g., Priest 1982). This model combines fluid equations and Maxwell's equations. See Sec. 2.8. Although the MHD theory is the simplest self-consistent model describing the macroscopic behavior of the plasma, the full nonlinear equations are so complex that usually simplifications are necessary to yield tractable problems (e. g., Hamabata 1989). Therefore, many solutions require numerical treatment. Finite-volume methods are one of several different techniques available to solve the MHD equations. They are simple to implement, easily adaptable to complex geometries, and well suited to handle nonlinear terms. Like solutions of hydrodynamic and other hyperbolic equations presented in the former sections, solutions of MHD equations exhibit the tendency to form large gradients (e. g., shock waves) which are difficult for numerical modeling. The use of standard numerical schemes of second-order accuracy or higher (e. g., the Lax-Fredrichs method) generates spurious oscillations which destroy monotonicity of the solution. Lower-order schemes (e. g., Godunov 1959) are generally free of oscillations, but they are so dissipative as to wash out much of the details. Therefore, there is a need to develop more advanced schemes which would adequately represent the large gradient profiles.

165

166

10.1

Numerical methods for the MHD

equations

Problems with the M H D equations

Converting an Euler code to a MHD code is not a straightforward task since various kinds of singularities are present in the MHD equations. Moreover, due to the intrinsic complexity of the MHD equations, the development of numerical techniques to solve these equations has been slower than for hydrodynamics (HD). For instance, for a long time most numerical schemes have been based on methods dependent on artificial viscosity to represent adequately shocks (e. g., Stone, Norman 1992). Although these schemes have been used successfully in astrophysical applications (e. g., Murawski, Steinolfson 1996a,b), the past experience with fully conservative, high-order upwind hydrodynamic codes found those to be superior in many applications (Woodward, Colella 1984). It is therefore natural to extend such schemes to solve MHD conservation equations. However, there are two principal difficulties associated with the numerical solution of the MHD equations as compared to the hydrodynamic (HD) equations (Roe, Balsara 1996). The first difficulty is that MHD equations possess new families of waves. Moreover, MHD admits a variety of exotic wave structures such as switch-on fast shocks, switch-off fast rarefactions, switch-off slow shocks, and switch-on slow rarefactions. It is also possible to obtain compound waves of either fast or slow waves. This has a considerable impact on a performance of the algorithms which are required to provide the stable and accurate capture of this entire range of such structures (e. g., Barmin, Kulikovskiy, Pogorelov 1996). Roe and Balsara (1996) list the six cases that can potentially cause trouble. The other difficulty is that the MHD equations contain the magnetic field which has to satisfy the divergence-free constraint, V • B = 0. A local nonzero divergence of magnetic field indicates the existence of magnetic monopoles within the numerical cell, which leads to nonconservation of the magnetic flux across its surface. Accumulation of the numerical errors associated with evolving the magnetic field components can lead to violation of this constraint, causing an artificial force parallel to the magnetic field, and eventually can force termination of the simulations. Despite of these problems many numerical schemes were developed for the MHD equations. These schemes reveal either conservative or nonconservative properties of the equations.

Conservative form of the MHD equations

10.2

167

Conservative form of the M H D equations

The MHD equations can be written in the conservative form u, t + V - f = 0,

V B = 0,

(10.1)

where: u=(g,gv,B,E)T,

(10.2)

B2 f = (gv, gvv + I(p + — ) - B B , v B - Bv,

(10.3)

(E + p+^-)v-B(vB))T.

(10.4)

Here I is the 3 x 3 identity matrix, vv stands for the 3 x 3 tensor v^j, and B has been normalized by y/ji. The momentum equation of (10.1) can be rewritten as follows: (0v),i + V • (gvv) + V(p + ^ ) - (B • V)B - B(V • B) = 0.

(10.5)

The last term of this equation should be equal to zero. If nevertheless V • B differs from zero, it becomes an additional unphysical force which is parallel to B . This force has a destabilizing effect on numerical algorithms. Brackbill and Barnes (1980) noted that this instability can be removed by adding the term - B ( V • B) to the right hand side of Eq. (10.5). This procedure leads to a non-conservative form of the MHD equations. In the finite volume method the plasma state of Eq. (10.2) is advanced in time by evaluating the fluxes of Eq. (10.4) at the interfaces between neighbouring states. In order for the Rankine-Hugoniot conditions to be satisfied at these interfaces, these fluxes must contain some kind of dissipation and a flux limiter must be applied to minimize post-shock oscillation. To eliminate these oscillations, a spacially averaged primitive state, u=(§,v,B,E)T

(10.6)

is required at the interfaces. Brio and Wu (1988) concluded that such an averaging is possible only for the case of 7 = 2. The averages for the mass density, velocity, pressure, and magnetic field are given by equations (8.67) and (8.68).

168

Numerical methods for the MHD

10.3

equations

Non-conservative equations

The MHD equations can be written in the non-conservative form (Powell 1994) u t + V-f = - V - B ( 0 , B , v , v B ) T ,

(10.7)

V - B = 0.

It is interesting to check how this change effects the induction equation which can be now written as B, t + v(V • B) + B(V • v) - (B • V)v = 0.

(10.8)

Taking the divergence of both sides and using the mass continuity equation, we obtain an advection equation for the quantity V • B/#, i. e.

(™)

+ vV(

(10.9)

] =0.

,t

As a consequence of that we have introduced a new divergence wave which propagates with the speed v. So, a partially conservative form of the multidimensional equations, obtained by adding terms proportional to V • B , retains the one-dimensional eigen-value problem, with the addition of an eighth wave that convects V • B as a passive scalar. The original MHD equations can be written in the quasilinear form u )( + Au | j ; = 0,

(10.10)

where: u=

f vx

A =

0 0 0 0 0 0

Io

Q

vx 0 0 0 By

Bz IP

0 0 vx 0 0 —Bx 0 0

0 0 0 vx 0 0 —Bx 0

(Q,QV,B,P)T,

(10.11)

0

0

Bx

5M. Q

B* Q

e

0 ~Vy -Vz

(7 - l ) v • B

0 0 vx 0 0

0

0 \

Q

e

0

0 0 0 0 0

1

B„ Q

0 0 vx 0

Vx

(10.12)

Non-conservative

equations

169

It is noteworthy that the 5-th row from the top of the matrix A consists of zeroes. This is a consequence of the fact that (V • B), t = 0. As a result of the zero row, we find that the 8-th eigenvalue of A is zero, i. e. w* ,t + 0ws >x = 0,

(10.13)

where w8 = Bx. Equation (10.7) for u = u can be written as /0 0 0 0 0 0 0 0

,< + Au, x = -

0 0 0 0 0 0 0 0

u

0 0 0 0 0 0 0 0 0 \0 0 0

0 0 0 0

0

0 0 0 LQ 0 0 vx Vy 0 0 Vz (7 - l)v • B 0

0 0 0 0 0 0 0 0

0\ 0 0 0 u, Bj 0 0 0 0/

(10.14)

where A is the matrix defined by Eq. (10.11). This equation can be rewritten in the quasilinear form u, t + Au,j = 0,

(10.15)

where: (

A =

Vx

Q

0 0 0 0 0 0

vx 0 0 0 By Bz IP

Io

0 0 vx 0 0 -Bx 0 0

0 0 0 Vx

0 0 —Bx 0

0 0 0 0 vx 0 0 0

0

0 EL Q

Q

0 0 vx 0 0

0 Q

0 0 vx 0

0 1.

a 0 0 0 0 0 vx

(10.16)

So, we see that the zero row has disappeared and the eight wave now satisfies the advection equation w6,t+vxwstX

= 0.

(10.17)

As this wave carries non-zero magnetic field divergence it is nicknamed the divergence wave.

Numerical methods for the MHD

170

10.4

equations

Eigenvalues and eigenvectors

The Jacobian matrix A has the eigenvalues (A) and left (1) and right (r) eigenvectors which correspond to the following waves (Powell 1994): Four magnetosonic waves with: A± = vx

/

0

/

\

±QC± ^ ¥

l ± = iV ±

BXByQC± Qci-Bl

e

±c±

\

BmByc±

^ecl-Bl

XQ^-Bl

B*Bzc± ^Q^-Bl

r±=N±

0 ByQc\

ByQc\

eci-Bl

B,QC%

B,gc% Qci-Bl

\

/

(10.19)

o

ecl-Bi eci-B* 1

\

(10.18)

±c±,

IP

J

where N^ stands for a normalization factor such that Y^r^ = 1. This factor is too complicated to be printed here. The superscript ± corresponds to the fast (c + ) and slow (c_) magnetosonic wave speeds which are given by Eq. (3.18); Two Alfven waves with: A°

=

vx±VA, 1

1°

=

2VW(0,0,-B„B„0,±^,^,0),

(10.21)

r

=

- ^ = ( 0 , 0 , -Bz, By, 0, ±BZJ~Q, TByJ-Q,0)T,

(10.22)

(10.20)

where N = l / ( 5 y + B\) is a normalized factor and the Alfven speed VA = Bx/y/Q;

One entropy wave with: Ae

=

vx,

(10.23)

V

=

-1, (1,0,0,0,0,0,0,—),

(10.24)

re

=

(1,0,0,0,0,0,0,0) T .

(10.25)

Singularities

171

Here the entropy s is defined as s = log (j^j

.

(10.26)

One divergence wave with: \div div

\

div

r

10.5

=

vx,

(10.27)

=

(0,0,0,0,1,0,0,0),

(10.28)

=

T

(10.29)

(0,0,0,0,1,0,0,0) .

Singularities

The Alfven eigenvectors become singular when

B± = Jsf+B* -»• 0.

(10.30)

The magnetosonic eigenvectors are singular for c± —> V\, (?+ —>•
& = %••

(10-31)

Then, the Alfven eigenvectors can be written as follows: 10± = \ ( 0 , 0 , ^ , ^ , 0 , - ^ ^ , ^ ^ , 0 ) r 0 ± = (O,O,±0z,TPvA-Pzs/dsgn(Bx),i3yy/Qsgn(Bx),O)T.

,

(10.32) (10.33)

The singularities in the Alfven speed can be fixed by applying (Brio, Wu 1988)

lim py = lim & = 4=-

(10.34)

172

Numerical methods for the MUD

equations

An elegant way of implementing the above limit into a code is to set Py =

By + e Bx + e v T

e«

(10.35)

1.

Now, we define a_ =

(10.36)

a, =

c i — cr_

A lengthy algebra leads to the eigenvectors for the magnetosonic waves (Abeele 1995)

/

° ±a+c+ Ta-c-/3y sgn(Bx) •^•a.-c-.$z sgn(Bx) 0 a-csPyl^Q

1+ = — 2cl

(

\ »+ -

Ot-Cs/3z/y/Q

V

<*+/e

J

0

>

(

<x+Q

±a+c+ Ta~c-/3y sgn(Bx) ^a-c-/3z sgn(Bx) 0 a-csPyy/Q a-cspzy/g

\

(10.37)

a+QC2s

(

OL-Q

±.CL-C-

V =

±a+c+f}y ±a+c+/3z

2c?

sgn(Bx) sgn(Bx)

,r

0 -a+CsPy/y/g -a+CsPz/y/Q \

a-lQ

-

±a+c+Pysgn(Bx) ±a+c+pz sgn(Bx) 0 -a+cspy^Q -a+CsPz^/e

(10.38)

J

These eigenvectors contain only the singularity which is called a triple umbilic T. It occurs at c\ = c2_ = V\ when B±, -> 0. The triple umbilic point is, where the fast, slow, and Alfven speeds coincide. It can be shown that around this point = cos — + d_

a+ = s i na- + 5+,

(10.39)

where: tana

_ Bx - c3y/g

(10.40)

Problems with MHD Riemann

solver

173

The errors 6± satisfy

ws d ^ For B± = 0 it can be proven that a_ = H(Bx—csy/g) Bx), where H is the Heaviside function. 10.6

<10-41) and a+ = H(cs^g —

Problems with M H D Riemann solver

An important problem in developing a scheme for MHD equations is that these equations are neither strictly hyperbolic nor strictly convex (e. g., Brio, Wu 1988). The MHD equations form a non-strictly hyperbolic system as some eigenvalues may coincide at some points and that compound wave structures, involving both shocks and rarefactions, may sometimes develop. It occurs that when the magnetic field components are equal to zero the eigenvectors become singular. By renormalizing the eigenvectors, these singularities can be removed (Brio, Wu 1988, Roe, Balsara 1996). See also the comment at the end of Sec. 10.5. Contrary to the hydrodynamic case, the Riemann problem for ideal MHD is not completely consistent and unique as one of the eigenvalues of the Jacobian matrix is zero. See Eq. (10.13). This zero eigenvalue is nonphysical as the eigenvalues should appear either singly as the x-component of the flow, vx, or in pairs symmetric about vx. Physical eigenvalues are given by Eqs. (10.18)^(10.29). The zero eigenvalue leads to numerical difficulties associated with nonzero divergence of the magnetic field. Consequently, characteristics can become degenerate, depending on the orientation of the magnetic field. It turns out that the solution of this problem is to consider a form of the equations that is not strictly in a conservation form (Powell 1994). See Eq. (10.7). 10.7

Divergence cleaning schemes

There are several important issues in developing a new MHD code. One of these is ensuring V • B = 0 (c. g., Dellar 2001). It is well known that incorrect treatment of the induction equation will lead to a non-selenoidal field that varies in time and hence causes the magnetic field to exert a

174

Numerical methods for the MHD equations

non-physical force along field lines. It occurs that the discretization errors lead to non-zero divergence over time. Physically, this means that nothing maintains conservation of a magnetic flux in the Gauss' law. This error usually grows exponentially during the computations, causing an artificial force parallel to the magnetic field, unphysical plasma transport orthogonal to B as well as a loss of momentum and energy conservation, destroying the correctness of the solutions (Brackbill, Barnes 1980). Several remedies have been proposed. Brackbill and Barnes (1980) found that the momentum equation can be reformulated into a non-conservative form to eliminate the parallel force. Harder (1987) proposed a method of adding a diffusion term in the induction equation that makes the divergence-free error diffuse away from the source. By this term magnetic monopoles are locally suppressed but they are not completely eliminated. Ewans and Hawley (1988) utilized a numerical technique called constraint transport to transform the induction equation in such a way that it maintains vanishing divergence of the field components to within machine round-off error by placing field components at appropriate locations of a numerical cell. This technique was used by Stone and Norman (1992) who implemented a covariant formalism in the ZEUS code which is based on the method of finite-differences. In another constraint transport method, a magnetic field is kept divergence-free to within machine round-off error by placing the magnetic field components at the interface locations of the finite-difference grid (DeVore 1991). Until recently, there were four traditional approaches to enforce the divergence-free constraint: a) a magnetic vector potential, B = V x A, approach. Then, the divergence-free condition is satisfied automatically. The difficulty with this approach is that a representation of the Lorentz force requires taking a second derivative of the vector potential A. That forces an application of the higher-order numerical schemes. Even then, one can encounter serious problems due to anomalous Lorentz force which apparently reveals itself in the neighborhood of large gradients; b) a projection scheme which forces the divergence-free constraint by solving a Poisson equation to subtract off the portion of the magnetic field that leads to non-zero divergence (e. g., Tanaka 1993b). The essence of this method is as follows. Suppose that magnetic field has a non-zero divergence, V ' B ^ O . We can fix this problem by adding a correction term B c such

Divergence cleaning schemes

175

that V • (B + B c ) = 0.

(10.42)

Clearly, B c must not generate new current j c = i ( V x B c ) = 0. Hence, V x B c = 0,

(10.43)

B c = V0,

(10.44)

from which we conclude that

where <j> is a scalar potential. Substituting Eq. (10.44) into equation (10.42) we obtain V2 = - V - B .

(10.45)

This is the Poisson equation which has to be solved in the whole computational domain. The resulting solution (f> should be used to evaluate B c according to Eq. (10.44) and this to clean the magnetic field B . This method has its disadvantages. Its major drawback is that it requires a global solution to the elliptic Eq. (10.45) which is computationally expensive. Moreover, the global nature of the cleaning procedure violates the hyperbolicity of the MHD equations in regions where the flow is supersonic and superalfvenic; c) a staggered-grid approach in which the divergence-free constraint is satisfied by placing the magnetic field components at the centroids of appropriate cell faces and volumetric variables such as mass, momentum and energy are stored at the centroids of computational cells. On such a grid the MHD equations can be approximated in a way that preserves selenoidality of discrete magnetic field (DeVore 1991, Stone, Norman 1992). This approach comes from incompressible fluid mechanics where the velocity field must be kept divergence-free. Staggered grids are expensive for storage and handling on meshes with hanging nodes that are common to unstructured grid methods. Moreover, appropriate Riemann solvers do not seem to work on staggered grids; d) the truncation-level error method which has been developed by Powell (1994). See also Gombosi et al. (1994), Asian (1999), and Powell et al. (1999). That approach relies on an addition to the original set of MHD equations the source term that is proportional to V • B. See Eq. (10.7). By that way any local V • B that is created is convected away in accordance

176

Numerical methods for the MHD

equations

to Eq. (10.9). That approach leads to the Riemann problem which has an eight-wave structure, where seven of the waves are those used in previous works on upwind methods (e. g., Huynh 1995) for MHD, and the eight wave is associated with the divergence of the magnetic field. It has been found by Janhunen (2000) that in the case when the contribution to the total energy from the fluid pressure is small in comparison to the magnetic and kinetic energies this approach may lead to an unphysical intermediate state with negative fluid pressure. As a consequence of that computing the pressure from the conserved quantities may involve the difference between two nearly equal terms and the errors result. Janhunen (2000) showed that this problem can be overcome by discarding the source terms in the energy and momentum equations, so that Eq. (10.7) becomes ut-|-V-f = -V-B(0,0,v,0)T,

V - B = 0.

(10.46)

This equation has been derived from relativistic energy-momentum conservation by Dellar (2001). The question of divergence cleaning in a rather detailed way is taken up by Balsara (1994, 2001). Essentially it boils down to saying that if the equations were exactly solved in a discrete fluid dynamic code the divergence would stay zero if it were zero initially. In reality one makes some small errors every time-step. It is only after a lot of time-steps that the errors build up. (This happens in a 7-th or 8-th wave model.) Thus the cure of removing the divergence only needs to be applied once in a while. Numerical experiments comparing various schemes with respect to the V • B = 0 constraint were performed recently by Toth (2000) who showed that the truncation-level error method performs generally well. However, in strongly discontinuous and stagnated flows this method accumulates so much magnetic monopoles that they corrupt the solution.

10.8

A scheme for a strong magnetic field

In the case of a strong magnetic field it is useful to expand the full magnetic field B as the sum of the background field B 0 and a perturbation Bx, i. e. B = B0 + Bi.

(10.47)

A scheme for a strong magnetic field

177

We assume that B 0 satisfies the following constraints: B 0 , t = 0,

(10.48)

V • B 0 = 0,

(10.49)

V x B 0 = 0.

(10.50)

We define now the perturbed state u1 = (0,0v,B1,£1)T,

(10.51)

where

£

+

+

'=^T 4 f

10 52

<'>

is the perturbed total energy density. In terms of the state ui Eq. (10.7) can be rewritten as follows (Tanaka 1994, Powell et al. 1999): u i , + V • fi + V • g = - V • B ! ( 0 , B , v, v • B X )

(10.53)

with V • B j = 0. Here B2

fi =(0v,0vv + I ( p + - ^ - ) - B 1 B 1 , v B 1 - B 1 v , (E1+p+^)v-B1(vB1))T,

(10.54)

g = (0, (Bo • B i ) I , (BoBx + BiBo), v B 0 - B 0 v, (Bo-B1)v-(vB1)B0)T.

(10.55)

This scheme has exactly the same eigenvalues and eigenvectors as in Sec. 10.3. However, the Jacobian matrix is different (Powell et al. 1999). The accuracy of this scheme is improved due to the expansion of the full field B , given by Eq. (10.47).

178

10.9

Numerical methods for the MHD

equations

Few specific examples of explicit M H D schemes

In this part of the book we present several explicit MHD codes which were developed recently. We limit ourself to few examples which were not mentioned above. Zachary and Colella (1992) used the scheme which estimates fluxes in MHD conservation equations. In another approach, Zachary, Malagoli, and Colella (1994) developed a multidimensional code for ideal magnetohydrodynamics. This code is higher-order Godunov method which adopts the piecewise linear interpolation, followed by projection onto characteristics for the construction of the left and right states at the cell boundaries. A multi-dimensional numerical code for solving the ideal MHD equations has been presented by Ryu and Jones (1995). This code is a secondorder-accurate extension of the Roe-type upwind scheme (Roe 1981). The constraint of a divergence-free magnetic field is enforced by calculating a correction via a gauge transformation in each time-step (Brackbill, Barnes 1980). Multiple spatial dimensions are treated through a Strang-type operator splitting (Strang 1968). The Riemann solver which was applied by Ryu and Jones (1995) is similar in many aspects to that developed by Dai and Woodward (1994a,b). It treats rarefactions properly, instead of approximating them as rarefaction shocks as was done by Dai and Woodward (1994a,b) who included rarefaction waves in their MHD Riemann solver by assuming these waves could also be treated as discontinuities. So long as the rarefactions are weak this is reasonably accurate but not exact. By conserving all Riemann invariants through the rarefaction waves, one can derive a simple set of differential equations to be integrated through the rarefaction waves (Ryu, Jones 1995). Balsara (1994) has constructed a TVD scheme for solving the onedimensional equations of ideal MHD. Piecewise linear interpolation has been applied to the plasma variables along with steepening of linearly degenerate characteristic fields i. e. the entropy wave and the two Alfven waves. An artificial viscosity and hyperviscosity have been formulated to remove numerically induced oscillations. This viscosity can be useful in the case when a very strong magnetosonic shock moves very slowly past the computational grid. As a consequence of a discreteness of the grid small amplitude post-shock oscillations will be generated. Artificial hyperviscosity (higher-order viscosity) may be also desirable in turbulence simulations and when one wants to iterate the solution to steady state. As a conse-

Few specific examples of explicit MHD

schemes

179

quence of a presence of the slow magnetosonic shocks, the artificial viscosity needs to be considered very carefully. Thus the wave speeds that are used in the construction of such a viscosity should adjust to the character of the wave being filtered up. The viscosity should leave the linearly degenerate characteristic fields unchanged as much as possible. A second-order Riemann solver for multi-dimensional MHD equations was proposed recently by Dai and Woodward (1998). In this scheme the divergence-free condition was exactly satisfied (although the scheme does not involve any Poisson equation) by setting the magnetic field components at interfaces of grid cells. AMR-MHD scheme for the second-order accurate, divergence-free evolution of a magnetic field on an adaptive mesh refinement hierarchy was developed by Balsara (2001). Both the two- and three-dimensional reconstruction strategies were worked out with high level of paralelisation. The other codes which are worth mentioning are: a pseudo-spectral fast Fourier transform algorithm for the solution of the three-dimensional resistive MHD equations in toroidal geometry was developed by Schnack, Baxter, and Caramana (1984). Three-dimensional MHD numerical scheme for two-component reacting plasma was developed by Tanaka and Murawski (1997). Two-fluid three-dimensional MHD numerical algorithm, based on ZEUS code, was constructed by Jun, Clarke, and Norman (1994). A Roe scheme for ideal MHD equations in adaptively refined triangular grids was devised by Peyrard and Villedieu (1999). 10.9.1

9-th wave Riemann equations

solver for two-component

MHD

To discuss the 9-th wave Riemann solver for two-component MHD equations it is useful to denote densities of two-component plasma by Qi and g2. Then, the total density is Q = Qi + Qi-

The background potential field and source term are expressed by B 0 and S, respectively. Introducing new dependent variables ui = (g,m,Bi,Ei,Q2)T Bx - B0x,By

- B0y,B2

- B0z,E-

-

(Q,mx,my,mz,

(Bi-B 0 )//i - Bg/(2ji), Q2)T, (10.56)

Numerical methods for the MHD equations

180

with the rotation of dependent variables u n = Tu = (g, m„, B„, E, g2)T = (g, mn, mti, m t 2 , Bn, Bn, Bt2,E,

g2)T,

(10.57)

the equation for ui can be written as (Tanaka, Murawski 1997) ^ J vndv + f T - 1 F ( u l n , B o n ) d s = f Sdv,

(10.58)

where dv and ds are the volume and surface elements of the control volume and T is a matrix which rotates the x-axis to the direction of a unit vector n normal to the surface of the control volume. The flux function F in the normalized form is written as mn P+ ^

+f ^f^e TO 2 m 'e "

F =

- \BnBn - § + ±B0nB0n -^BtltB \Bnn + -BotiBo - ±B j;BotlB0nn — -^BtiBn 0

+ -Bot2-Bo ?0n

?fBt2 - sfBn m •(E! Q

(10.59)

+

x(^Bln + ^Bltl + ^Blt2) + S^(!B?Botl - ^Bon) £2.™

Eigenvalues and eigenvectors For the Jacobian matrix of 9-th component flux function, eigenvalues A" m = 1 , . . . ,9 are A1 A2,3 A4,5

6 7

A-

8

A

9

A

(10.60)

=

m

=

<±\K

(10.61)

=

m'n±c+,

(10.62)

=

m'n±c-,

(10.63)

=

o,

(10.64)

=

m',

(10.65)

'n>

Few specific examples of explicit MHD schemes

181

with the notation

u'n = ( ^ m ' n , B ^ E , g 2 ) T =

{g, mn/g, mtl/g, mt2jg, Bn/y/JIg, Ba/y/JIg, Ba/^/iiQ,

E, g2f

Here, variables with ' have a dimension of velocity and \B'n\, c+ and c_ correspond to Alfven, fast and slow speeds, respectively. The eigenvectors rTO which correspond to Am are

1 m'n m!t\ ,1

,.2,3

-

(10.66)

0 0 0 m' 2 /2 0

0 0 ^Bksgn{B'n) ±B't'1-sgn(Bn) 0

=

^{B'^m'n

- B't[m't2ysgn(B'n) 0

+ {B&B'ltl

-

B't[B[t2) (10.67)

Numerical methods for the MHD

equations

af af(m'n±c+) afm'nTasB^c+B'n afm't2TasB't'2c+B'n 0 (10.68) a,f0.5-m'2 + a,fC+/(y — \)±a,fC+m'n ^asc+{B^m'tl + B'{2m't2)B'n

+a / (-l)/( 7 -l)(4-c 0 ) +af(cl-co)(B't>1B{'tl+B>2Bl>t2) >n 2 /(Btf + B'tf) t2 f

aSQ2JQ

as(m'n±c-) asm'tl±afB't'iy/cS/c+-sgn(B'n) asm't2±afB?2^cS/c+-sgn(B'n) 0

-afB^co/cl -afB'^colc\ as-0.5-m'2 + a s c?_/(7 l)±asC-m'n ±aj(B,^m'tl+B'l2m't2) x^cS/c+-sgn(B'n) + o , ( - l ) / ( 7 - \){c2_ - c0) +a s (c 2 _ - c o X i ^ ' t i + S^JB/^) /(B'A 2 +B'/ 2 2 )

(10.69)

Few specific examples of explicit MHD schemes

' 0 ' 0 0 0 1 , 0 0 0 . 0 .

183

1 < "*ii m

r9 =

't2

0 0 0 0.5-m'2 1

where: B'A = (B'a + e)/(B'tl + B'i + 2e 2 ) 1 /2,

(i 0 .7l)

B'k = (B't2 + e)/(B'tl + B'tl + 2e2y/\

(10.72)

O/ = ( 4 - B ;

3

)

1 / 2

/(4-0

1 / 2

,

as = ( 4 - c 0 ) 1 / 2 / ( 4 - c 2 _) 1/2 c + .

(10-73)

(10.74)

The symbol e is a small number and CQ is the sound speed. Then the upwind numerical flux F^- at the interface of control volumes i and j can be written as Etj

=

p [ F ( u i n j , B o n i ) + F(ui„i,Borai) ~ R j j | Ajj | R ^ ^ U i n j - Ui„i)],

(10.75)

where the eigenvector matrix Rjj and the eigenvalue matrix A^ are calculated from the symmetric average of ui nj - and u i n j . To get a higher-order accuracy, the MUSCL approach is used with indices i and j being replaced by r and I, suffixes which indicate variables just on the left and right sides of the interface (Tanaka 1994). It is noteworthy that a numerical scheme has been developed by Shyue (2001) to model multicomponent fluids with the general Mie-Gruneisen equation of state. This scheme combines the Euler's equations of gas dynamics with a set of effective equations for the material-dependent functions.

184

10.10

Numerical methods for the MHD equations

Implicit M H D schemes

There are three approaches in solving MHD equations implicitly: fully implicit scheme, semi-implicit scheme, and barely implicit scheme. While a fully implicit scheme treats all the terms implicitly, a semi-implicit approach adds some small additional terms to the MHD equations that vanish as the time-step goes to zero (Harned, Kerner 1985). That method has been demonstrated to work reliably well (Harned, Schnack 1986, Kerner 1990, Klein 1995). The semi-implicit schemes have the advantage of not requiring iterations, but inaccuracies are introduced by the operator splitting that limit the allowable time-step. An alternative approach is less compute-intensive. It follows a technique, called barely implicit correction, that was developed by Patnaik et al. (1987) for slow-flow hydrodynamic problems. See Sec. 9.5.1. A similar approach is also possible in the case of MHD equations (DeVore 1996). However, trans-Alfvenic flows will not benefit from a barely implicit technique, which is only useful for slow flows. In that situation, a fully implicit method is the only way to go. There were a few attempts of developing implicit MHD codes. In particular, the fluid-implicit-particle method (FLIP), was extended to magnetohydrodynamic flow in two or three dimensions by Brackbill (1991). This method incorporates a Lagrangian representation of the field and is shown to preserve conservative properties of plasma as well as contact discontinuities and the Galilean invariance of the MHD equations. Another implicit MHD scheme was developed by Hu (1989). The author used the alternating direction iteration (ADI) scheme. This scheme was implemented for solving two-dimensional MHD equations. An implicit numerical code for the magnetospheric plasma was developed by Rankin, Samson, Frycz (1993). This code uses the Douglas-Gunn algorithm (Douglas, Gunn 1964) for ADI temporal advancement, with an overall second-order accuracy in both space and time. The implementation of the ADI allows the use of a time-step that is many times larger than the time-step in any explicit code. In the three-dimensional case, the algorithm proceeds by advancing the system of MHD equations along one spatial direction at a time, which involves the iterative solution of nonlinear block tridiagonal systems of algebraic equations. A new implicit scheme for solving the non-ideal MHD equations was presented by Jones et al. (1997). This scheme encompasses a finite-volume method that uses an approximate Riemann solver. The implicit terms are

Implicit MHD

schemes

185

inverted using a lower-upper symmetric Gauss-Seidel iteration in conjunction with dual time-stepping to maintain time accuracy. Most of these codes have been found to be ineffective for strong magnetic field plasma application as there was not any significant improvement of the CPU time (Schnack et al. 1990). That was mainly due to the fact that for three-dimensional MHD problems, inverting the full implicit operators became impractical. The problem with implicit schemes is that there is only a relatively narrow range of conditions for which they have a clear computational advantage, and even then one can always wonder if too much of the physics was numerically filtered out to get the right answer. That, of course, depends on the types of problems one wants to solve. As it is well known, there is not one type of code that is best for all problems.

Chapter 11

Numerical experiments

As far as spatial and temporal scales are involved, fluid simulation methods can be divided into three groups: (a) magnetohydrodynamic (MHD) simulations are associated with large spatial L and temporal r scales (e. g., Ogino 1993, Tanaka 2000a); (b) hybrid simulations which imply kinetic ion and massless fluid electron. They are associated with middle scales (Winske, Omidi 1993, Brecht, Ferrante 1991); (c) particle simulations correspond to small scales (Tanaka 1993a, Shimazu et al. 1996). Here, r* is the ion gyroradius, Ui is the cyclotron frequency, tjpe is the electron plasma frequency, and c is the speed of light. For space plasma simulations these scales are respectively L > ri ~ 100 km and r > 1/WJ ~ 10 s for (a), L ~ 100 km and r ~ 10 s for (b), L ~ c/ujpe ~ 1 km and r = l/w p e ~ 10~ 5 s for (c). We limit here to the discussion of essentially the first group of simulations although the ion-acoustic waves simulations, which belong to the second group, are also briefly considered. The choice of examples presented here is somewhat subjective. Rather than choosing representative problems from all areas of fluid physics, we have primarily concentrated on specific topics on which we have done a considerable amount of work. These topics are presented below.

187

188

11.1

Numerical experiments

Numerical solution of the inviscid Burgers equations

In this part of the book we present numerical solutions to equations (4.19) and (4.23) which are rewritten here in the compact form Vz+(l

+ pnVn)vt

= 0,

(11.1)

where J3n is the normalized nonlinear coefficient and n = 1,2 correspond to Eqs. (4.23) and (4.19), respectively. Numerical solutions to this equation are obtained by use of the fast Fourier transform method, which is now briefly described. The idea of the fast Fourier transform method is to calculate the temporal derivative of the unknown, V, with the following scheme: V,t = T-1{imT{V{z,t)}},

(11.2)

where T~x and T are the inverse Fourier transform and Fourier transform operators, respectively, and m denotes a Fourier mode number. The spatial derivative is discretized using the centered Euler scheme,

_V(z +

Az)-V(z-Az)

Now Eq. (11.1) can be discretized in space and time. This numerical scheme has been checked to be stable and found very accurate. Numerical results for Eq. (11.1) with n = 2, i. e. the model equation for Vx, are displayed in Fig. 11.1, which shows the signal measured at z = 0.5 L as a function of time, and the corresponding Fourier spectrum (Fig. 11.1b). Here L is a spatial scale. The driving signal at z = 0 is given by Vx(z = 0,t) = Vosm(ujdt).

(11.4)

It is shown in Fig. 11.1a as a broken line for comparison with the detected signal. Here, the amplitude Vo = 0.3 VA and driving frequency Ud — 4n VA/L have been chosen. This relatively large amplitude is used to enhance the influence of nonlinearities on the velocity. Such as predicted by Eq. (4.20), the most obvious effect of the cubic nonlinear term in the model equation for Vx is to speed-up both positive and negative perturbations in such a way that the propagation speed grows with the absolute value of the velocity. As a consequence of that, peaks will never catch-up with valleys and both will move quicker than nodes.

Numerical solution of the inviscid Burgers equations

189

•§ 4

I*u

<

i 3a

t

Q)

i

I' 0.2

0.4

0.8

40

tvyL

S 0.15

d

* 7
I

0. 0

0.0

Sa

eo uL/VA

l2ut

•

~S o.to u

\4ut

* O.OS

lBud

«

lBud

ft, 0.00 30

SO

90

120

uL/Vt

Fig. 11.1 Top-left panel: Numerical solution to the inviscid Burgers Eq. (11.1) with n = 2. The solid line represents the signal detected at z = 0.5 L displaced —0.5 L/VA in time so that the disturbance is only shown from its time of arrival to the detection point. The broken line corresponds to the driving signal at z = 0 (Eq. (11.4)). The driving frequency and amplitude are u>d = 47r VA/L and Vb = 0.3 VA- Top-right panel: MEM spectral estimation of the detected signal in (a) using 410 MEM coefficients. Peaks can be seen at frequencies uid, 3ud, 5u>d, and 7u>d, indicated by arrows. Bottom-left panel: Same as (a) but with n — 1 and driver given by Eq. (11.5). Bottom-right panel: MEM spectral estimation of the detected signal in (c) using 365 MEM coefficients. Peaks can be seen at frequencies 2w
T h u s , t h e measured signal resembles saw-tooth oscillations, clearly seen in Fig. 11.1a. Wave breaking can occur at a later time at t h e leading (left) sides of b o t h humps and valleys, contrary t o what is observed in water waves breaking on a beach where t h e wave breaking occurs at t h e leading sides of humps and trailing sides of valleys. In addition, one of t h e conclusions from section 4.1.1 is t h a t t h e nonlinear effects in Vx are of odd order (third, fifth, etc.). W i t h t h e sinusoidal driver (11.4), cubic t e r m s are equal t o a sum of trigonometric functions with arguments iJa and 3cjd- In t h e same way, fifth order t e r m s contain periodic variations with frequencies LJd and 3w<j, and 5u>d, and so on for higher orders. Hence, it is expected t h a t t h e power spectrum of t h e signal

190

Numerical

experiments

measured at z = 0.5 L shows traces of power at frequencies (2n + l)ud with n = 0 , l , 2 , 3 , 4 , - - - and that higher order modes possess smaller amplitudes since they contribute less to the solution because they acquire less energy. Nevertheless, they are important since they are responsible for producing the steep profiles in Fig. 11.1a. We have therefore estimated the power content in this time series using the maximum entropy method (MEM) power spectrum estimation (Press et al. 1992). MEM spectral estimation is sometimes better than other spectral techniques (such as FFT) at defining the positions of peaks with substantial power, although it does not predict their power well: a change in the number of MEM coefficients can drastically change the value of the estimated power, but it will not introduce spurious peaks. MEM spectral estimation for Vx is shown in Fig. 11.1b. There is no doubt about the presence of power at odd multiples of the driving frequency in the signal. The above discussion refers to Vx. See Sec. 4.1. Oscillations in Vz are now excited by a driver described by

Vz(z = 0, t) = iy 0 2 sin2 (udt) = - Jy 0 2 [cos (2udt) - 1],

(11.5)

where VQ = 0.3 VA and Wd = 4TT VA/L have been used. The form of this driver is suggested by Eq. (4.22) with s = 1. The results of solving Eq. (11.1) with n = 1 are shown in Fig. 11.1c, where the solid and broken lines are the detected and driving velocities. As a consequence of the quadratic nonlinear term in Eq. (4.23), the positive signal is speeded-up and so the leading part (left part of waves in Fig. 11.1c) of the wave is steepened while the trailing part is smoothed out. Now, even multiples of ujd a r e expected to shape the detected signal. This is confirmed by its power spectrum, although leakage prevents from accurately determining the position of peaks. MEM spectral estimation does reveal the presence of power at the driving frequency 2u)d and at nonlinearly driven Fourier harmonics with frequencies 2nu)d, n = 2,3,4 (Fig. 11.Id). It is noteworthy that nonlinear effects on Fourier modes have also been observed by other authors (Allan, Manuel, Poulter 1991) in numerical experiments of impulsive waves in the Earth's magnetosphere.

The effect of random mass densityfieldson acoustic waves 11.2

191

The effect of random mass density fields on acoustic waves

In recent years, the propagation of waves in random media was the subject of intensive studies in the most diverse branches of physics. For instance, the effect of turbulence in the solar atmosphere on the surface-gravity wave amplitudes and frequencies was discussed by Murawski (2000b, c). In another context, Murawski (2000a) showed that a space-dependent random flow shifts the frequencies of acoustic waves and alters their amplitude. Fast magnetosonic waves which are impulsively generated in plasma, having a random mass density, were discussed by Murawski, Nakariakov, and Pelinovsky (2001). They showed that the localized pulses experience a spatial delay and attenuation due to the space-dependent (step-wise constant) random field. Attenuation of nonlinear waves was considered by Wadati (1990) who proved that a soliton, propagating a distance a; in a a;-dependent random medium, has its amplitude decreased as l/-\/^ a n < i its width increased as ^/x. Lipkens and Blanc-Benon (1995) showed that as a result of turbulence the nonlinear distortion of a pulse is weaker than in the deterministic case. We consider here a simple case of one-dimensional acoustic waves propagating in a fluid whose density is a random function of time. To provide a consistent and general treatment of wave propagation, we develop a numerical approach for which the deviations of the random density fluctuations from their mean value and the wave amplitude are not necessarily small. In this section, we present the results of the numerical simulations for Eqs. (2.34)-(2.36). These simulations are performed with a use of the CLAWPACK code (LeVeque 1997a,b,c), which is a packet of Fortran routines for solving hyperbolic equations. The code utilizes the wave propagation method (LeVeque 1997a and Sec. 8.1.3). Initially, at t = 0, the equilibrium state is set as follows:

0o = const, VQ = 0, po — const.

(H-6)

Here Qo,vo, and p0 are the background mass density, velocity, and pressure, respectively.

Numerical experiments

192

11.2.1

Seeding time-dependent

random

field

Random field is seed through the term Se in Eq. (2.34) such that Se = QTtt,

(11.7)

where:

sr(tm) =ftUjfT;

«n„)e<-*™*+'*»>) = V2NM {T-1 ( e ( n n ) e - * - ) ) .

(11.8)

Here N denotes number of Fourier modes taken into account, <j>n is a random phase with uniform distribution over the interval (0,2ir), chosen by the random number generator rani (Press et al. 1992). The quantity g{£ln) denotes the mode's amplitude e(n„) = y/E^j.

(11.9)

The spectrum E(U) is sampled uniformly by N values f2„ Zi'KTL

where: m

~

•

N

In Eq. (11.8) t is sampled as follows: tm = t0+mAt,

m = 0,1,2, • • • , i V - 1.

(11.10)

With the use of the correlation theorem (Press ei a/. 1992): >= \Qr(k)\2

T < Qr(tmi)Qr(tm2)

(11.11)

we find that the random field, which is generated by this way, possesses the required spectral properties. Fig. 11.2 shows a typical realization of the random field which was generated by the present method. A similar way of seeding the random field was developed by Juve et al. (1999). Waves are excited through the initial condition n

v(x,t = 0) =vQY^sm{ix), i=l

(U-12)

The effect of random mass density fields on acoustic

waves

193

where vo is the wave amplitude. So, this initial perturbance consists of n sinusoidal waves. The periodic boundary conditions are applied at the edges of the simulation region which are typically chosen at x = 0 and x ~ 82co/f.

0.2

T

I

10

20

I

r

0.1

<* o.o -0.1 -0.2 0

30

40

50

t

Fig. 11.2 Random mass density as a function of time t for a typical realization of random medium.

11.2.2

Numerical

results

In Fig. 3.1 the frequency of maximum spectral concentration is compared, for each wavenumber. The span of the error bars equals the standard deviation of the frequency shift over the random density statistical ensemble. We were able to observe frequencies which essentially agree with the theoretical prediction. A qualitative difference has been found for long waves (K ~ 1) for which numerical findings reveal wave deceleration and damping while the analytical results show frequency increase and wave amplification. Amongst the several sources of discrepancy between the numerical results and analytical predictions, a most important one is the breakdown of the statistical properties of the random wave operator which are necessary for the derivation of the random dispersion relation in Eq. (3.55).

Numerical experiments

194

11.2.3

Summary

In this part of the book we have presented a numerical study of the propagation of the sound waves in a medium with time-dependent random mass density fluctuations. The main findings can be summarized as follows. The time-dependent random mass density field leads to frequency increase and amplification of sound waves (Murawski, Nocera, M§drek 2001). One important consequence of a time-dependent random density field is the capability of waves to speed up and enhance their amplitude as they propagate, a possibility which is excluded in the case of a time-independent random density field (Howe 1971). A practical demonstration of this consequence is a positive value of the imaginary part of the root of the dispersion function, which is also a noteworthy result of our investigation.

11.3

Numerical simulations of air-pollutions

Dynamics of air-pollutions can be described by the following system of hydrodynamic equations: 0.« + V - t e v ) (0v),t + (gv • V)v P,t + V-{pv)

=

Se + SD,

(11.13)

=

- V p - ^ y + Sv,

(11-14)

=

(1 - 7 )pV • v + Sp>

(11.15)

where g is the mass density, v is the velocity, p is the gas pressure, g is the strength of the gravitational acceleration, y is a unit vector in the vertical direction y, t denotes the time and the adiabatic constant, expressed by the ratio of specific heats cp/cv, is 7. The x-axis lies in the horizontal direction. Henceforth, this discussion is limited to the two-dimensional case only with d/dy = 0. In Eq. (11.13), SD is a coefficient of diffusivity which for a constant diffusive coefficient D = 0.333 • 10~ 4 m 2 /s can be rewritten as follows: SD = -DV2g.

(11.16)

The other source terms Se, S v and Sp are given by the following expressions:

Se = vey—e~

'o ,

S v = Seveyy,

Sp = —e—rvey.

(H-17)

Numerical simulations of air-pollutions

195

Here ge and vey denote the mass density and the y-component of velocity of the emitted pollutions, respectively. The quantity XQ denotes the half-width of the emittor which is placed at y = yo = 10 xo and |x| < xo11.3.1

Numerical

model

We solve Eqs. (11.13)-(11.15) by the CLAWPACK code which is based on a wave propagation method (LeVeque 1997a). The boundaries of the simulation box are placed at x = ±10zo, V = 0, and y = 20 zo- The computational domain is divided into a uniform grid of 133 x 133 cells in the x- and y-directions. The top, left and right boundaries are in the present model entirely open. The bottom boundary corresponds to a rigid wall. Initially, at t = 0, density and pressure correspond to the isothermal atmosphere with To = 293 K. Their profiles can be obtained from Eqs. (3.29) and (3.35), replacing z by y. The vertical profiles of the atmosphere are presented in Fig. 11.3. The ground pressure is assumed to be po(y = 0) = 101325 Pa and the mass density Qo(y = 0) — 1.29 kg/m 3 . These values correspond to the sound speed c s = 333 m/s.

y

A

|

Po.Po

T0

1

£->

Fig. 11.3 Equilibrium profiles of the mass density, pressure and temperature for the case of the isothermal atmosphere. The air-pollutions are emitted with the speed vey = 15 m/s. They are assumed to be five times hotter than the surrounding air. The emission is determined by the source terms of the mass density, velocity and pressure in Eqs. (11.13) - (11.15).

196 11.3.2

Numerical experiments

Numerical

results and

discussion

We consider first the case of the emitted pollution which is equally dense to the initially static air, ge = g0(y = lOzo). The pollution expands itself horizontally at the initial stage of its temporal evolution (left panel of Fig. 11.4). This expansion leads to rarefied gas, making the occurrence of the buoyancy force which acts upwards, eventually overcoming the gravity force and lifting up the pollution. As a consequence of the Kelvin-Helmholtz instabilities vertices are generated (right panel of Fig. 11.4). These vortices make a spreading of pollutions more difficult. It is noteworthy that the scenario of spreading heavier pollutions (ge > Qo) is essentially similar to the above discussed case of the equal densities. The only difference is that sometime the gravity force prevails over the buoyancy force which drags the pollutions to the ground (Murawski, Michalczyk 2001).

Fig. 11.4 Spatial distribution of pollutions for the case of the emission density ge equal to the ambient air density go at the height y = 10 Xo, i. e. ge — go(y = ICteo). The contour plots of the mass density of pollutions at t = 1.5 s (left panel), and the velocity vectors (right panel). A lower density of the emitted pollution, i. e. ge = 0.75 Qo(y = 10:ro), results in formation of clouds which at time t = 3 s are at x ~ ± 5 xo and y ~ 10 a;o- These clouds are accompanied by the plume which at time t = 3 s is settled at x = 0 and y ~ 18 XQ (left panel of Fig. 11.5).

Numerical simulations of air-pollutions

197

P^ PV

10

-5

Fig. 11.5 Spatial distribution of pollutions for the case of lighter emission density ge than the ambient air density go(y = lOzo), i. e. ge = 0.75 go(y = ICteo). The mass density of pollutions at t = 3 s (left panel) and the velocity vectors at t = 9 s (right panel). Fig. 11.5 reveals that there is the energy cascade from larger to smaller vortices which at t = 9 s are well seen at y w 15 XQ and x ~ ±7xo- Such cascade is typical for a turbulent medium. The velocity vectors in the neighbourhood of the vertical axis x = 0 are directed vertically upwards as the pollution movement is not disturbed there. Now we discuss the case of wind which blows horizontally from the left boundary, x = xmin- We consider two cases: (a) y-independent wind speed, (y); (b) the wind speed which grows linearly with y. Fig. 11.6 (left panel) displays spatial mass density profiles for the constant wind speed vw = 2.5 m/s and diffusion D = 0.2 cm 2 /s. Despite of the small value of vw, a spatial distribution of pollution is considerably affected by the wind which entirely alters the initially vertical direction of the emission. The mass density is highest in the neighbourhood of the outlet from the emittor, x = 20xo, V = 25£o (left panel of Fig. 11.6). A number of clouds that exhibit various spatial scales are characteristic for mild wind. The vortices, which occur at the border between the plume and the ambient air, penetrate a region far away from the emittor. The mass density, which is associated with these vortices, is hardly diminished as a rotation reduces diffusion. Inside the cloud at x ~ 55 xo, y — 50 xo the rotation is absent and in t = 15 s the mass density attains the value close to the ambient air

Numerical experiments

198

rho+D

(Vw

= 2.5

m/s)

(t = 15

s)

80

rho (Vw = 10->15 m/s) (t = 20.0000 s)

\

I

I

~

i

i

i

r

:

:

:'

; o

-= g > :

Fig. 11.6 Spatial mass density profiles in the diffusion case for the constant wind speed vw = 2.5 m/s (top panel) and for the linear wind speed vw(y = 0) = 10 m/s (bottom panel).

density level. The case of linear wind speed is presented in Fig. 11.6 (right panel). We assume that the wind speed at the altitude y — 100 XQ above the surface of the Earth, vw(y = 100xo), is 50% higher than the speed at the ground level of the Earth, vw(y = 0). Specifically, we consider the case of vw(y = 0) = 10 m/s. Then, vw(y = 100XQ) = 15 m/s. Here, we see a general structure of the plume. The ripples occur at x > 90 #o. As a consequence of growing with the height wind speed the top ripples are more enhanced than the bottom ones. The plume is shifted up as a result of enhanced (due to linear wind speed) buoyancy force.

Driven MHD waves in the solar corona

11.4

199

Driven M H D waves in the solar corona

Observations of the solar corona have revealed the high complexity of magnetic field structures (e.
200

Numerical experiments

are now fairly well developed (e.
11.4.1

Physical

model

In the solar corona the plasma /3 = p/(B2/2(j,) is much smaller than unity. Consequently, gas pressure terms can be neglected and the coronal plasma can be described by the cold MHD equations with the viscous and resistive effects neglected. See equations (2.89)-(2.92). These equations are solved in a Cartesian coordinate system in which the x-axis is directed horizontally along the solar surface and the z-axis is vertical. In this section we study two-dimensional perturbations only, hence the y-coordinate becomes irrelevant (d/dy = 0). Additionally, we assume that the y-components of the flow (Vy) and magnetic field (By) are identically zero, which removes the Alfven wave from the system. Moreover, as the cold plasma approximation removes the slow wave in the linear regime, the system (2.89)-(2.92) then describes the fast magnetosonic waves nonlinearly coupled to slow mode perturbations. A coronal hole is here modeled as a slab of plasma threaded by a vertical magnetic field Bz. 11.4.2

Numerical solution coronal plasma

of MHD equations

for the

solar

The set of nonlinear MHD equations (2.89)-(2.92) has been solved with a two-dimensional resistive MHD code (DeVore 1991) in which resistivity and pressure terms have been set to zero. The code utilizes the FCT

Driven MHD waves in the solar corona

201

method of Sec. 6.11. See also Boris, Book (1973a,b), DeVore (1991), and Murawski, Goossens (1994a,b). This method yields accurate results near steep gradients and moving contact discontinuities by adding numerical diffusion and antidiffusion fluxes to the scheme. Dissipation is added only in small localized regions to eliminate short-wavelength oscillations generated by numerical dispersion, while leaving the longer-wavelength phenomena minimally affected. To achieve this the dissipation has to be nonlinear and implemented locally, at regions of high-gradients. The antidiffusion fluxes are 'corrected' to preserve the positivity and monotonicity of the profiles. For more detailed description of the FCT method see Sec. 6.11. The resulting computer code (DeVore 1991) is fourth-order accurate in space and second-order accurate in time, and the zero divergence of magnetic field is preserved to within machine-dependent round-off errors. An explicit predictor-corrector method has been applied for discretization in time. Somehow modified versions of this code were successfully applied to simulate the resonant absorption (Murawski et al. 1996) and phasemixing (Nakariakov, Roberts, Murawski 1997, 1998) of nonlinear Alfven waves. 11.4.3

Numerical

results

The boundaries of the simulation box are placed at x = ±L, z = 0, and z = 8L. The computational box is divided into a nonuniform grid with 100 cells in the x-direction and 400 cells in the z-direction. At z = 0 and z = 8L we use free boundary conditions with the first normal derivative at the point adjacent to the boundary set to zero. Free boundary conditions at x = ±L are employed in the case of a horizontally homogeneous driver, while rigid walls are implemented for a nonhomogeneous driver by setting Vx = 0 there. These boundary conditions have been found to work well. Tests have been performed with a larger number of grid points, 300 x 500, and no significant differences in the solutions have been found. This fact ensures grid convergence and that the accuracy of results is not compromised by the grid size. Therefore, we conclude that the solution has been adequately resolved by the original mesh of 100 x 400 points. In order to maintain high spatial resolution in particular regions, we employ a nonuniform grid in which the computational cells vary in size. The cell size changes gradually, so that neighboring cell dimensions are stretched by only a few tens percent per cell (typically 0.2%). Cell sizes

Numerical experiments

202

were set in such a way that the finest resolution is in the center of the simulation box. For this purpose the grid generation routine (GGEN) has been adapted from the ZEUS2D code (Stone, Norman 1992). Initially, at t = 0, all plasma parameters are set to the homogeneous state (Eq. (4.1)). The system is made to evolve by specifying the flow along the lower boundary, which in our model represents the base of the coronal hole. This flow is excited by a driver described by Vx(x,z = 0,t) =

VQCOB

(n^-x^

8(z)H(t)sin(cjdt),

(11.18)

where S(z) and H(t) are the Dirac delta and Heaviside functions, respectively. The driver begins to work at t = 0, exciting fast magnetosonic waves, and acts for the entire run. Now, we consider a coronal hole 200,000 km wide (such as the one in Hara et al. 1991), which corresponds to L = 105 km. The value of the Alfven speed is taken as VA = 1000 km s - 1 , so that the Alfven transit time is L/VA = 100 s. The driving frequency in Eq. (11.18) has been taken in the range WVA/L to ATTVA/L, so the driving period varies between 50 and 200 s. These values are similar to typical values that can be found in the literature (e. g., Parker 1992). Moreover, different heights of the detection point have been considered and we will concentrate on two of them, 200,000 km and 400,000 km (i. e., 2 L and 4 L) above the photosphere. Finally, driving amplitudes in the range 0.003 V A - 0 . 0 3 VA have been used, which in dimensional units corresponds to 3-30 km s _ 1 . The case of a homogeneous driver We first consider perturbations excited by a homogeneous photospheric driver, described by Eq. (11.18) with n = 0. Numerical results for the full set of MHD equations are displayed in Fig. 11.7a, in which Vx, Vz, and Bx are shown. The three signals have been measured at the point (x = 0, z — 2 L) as a function of time and, in order to enhance nonlinear effects, Vo is set to 0.3 VA- As a consequence of the cubic nonlinearity acting on the normal flow, Vx exhibits steepening and results in saw-tooth oscillations such as those found for the inviscid Burgers equation (compare solid lines in Figs. 11.1a and 11.7a). Fig. 11.7a also indicates that the parallel flow, Vz, is of lower magnitude and possesses twice higher frequency than Vx. This is in agreement with the analytical finding given by formula (4.22). In the full set of nonlinear MHD equations, Vz is nonlinearly driven by a quadratic term (c/. Eq. (4.7))

Driven MED waves in the solar corona

<£

203

0.020 0.010

<

0.000

o -0.010

<

-0.020

a. w 1.5

2.0

2.S tVj/L

3.0

3.5

0

20

40

60 aL/VA

80

100

Fig. 11.7 Numerical solution to the nonlinear MHD equations (2.89)-(2.92) for an x-independent driver (Eq. (11.18) with n — 0) with frequency uid — 4ir VA/L and amplitude Vb = 0.3 VA- (a) Time evolution of Vx (solid line), Vz (broken line), and Bx (dotted line) at x = 0, z = 2 L. Compare Vx and Vz with Figs. 11.1a and 11.1c. (b) MEM spectral estimation of Vx in (a) using 365 MEM coefficients. Compare with Fig. 11.1b.

which, given the form of the driver of Vx (Eq. (11.18)), is equivalent to driving Vz by a driver with a time dependence similar to Eq. (11.5). Hence, it is not surprising to find such a good agreement between the solid curve in Fig. 11.1c and the broken curve in Fig. 11.7a. Finally, comparison between the results for the horizontal component of the velocity and magnetic field shows that they oscillate in antiphase with the same amplitude. In fact, Vx and — Bx are visually indistinguishable. This is in agreement with the analytical progress done before. Indeed, using Bx ~ el/2BxX and Eq. (4.12) with s = 1 we obtain B'x = -V±,

(11.19)

where B'x = Bx/B0 is the dimensionless horizontal magnetic field component. We have again computed the power spectra and MEM spectral estimate of the signals to study their power content for a range of frequencies. It is found that Vx displays power at the driving frequency, uid, plus power (with decreasing amplitude) at odd multiples of LJd (see Fig. 11.7b). On the other hand, Vz contains power at even multiples of the driving frequency (not shown). These results are also in agreement with the weakly nonlinear study performed before.

204

Numerical experiments

The case of an inhomogeneous driver We present here results for driver (11.18) with n ^ 0. As the horizontal wave number is different from zero now, it follows from the dispersion relation for the fast magnetosonic waves (Cadez, Oliver, Ballester 1996),

where LOC = VAkx is the cut-off frequency, that fast magnetosonic waves are dispersive in the linear regime and there is a cut-off frequency (uic = n § VA/L) below which these waves are evanescent and cannot carry energy high into the corona. Let us consider in some detail the case n = 3 for which uic = 3f VA/L and let us take a driver with small amplitude, Vo = 0.003 VA, and oJd = Ait VA/L > u)c. After three driving periods, that is, at t = § L/VA, perturbations have traveled from the photosphere up to z = 1.5 L, showing no signs of vertical evanescence (see Figs. 11.8a and b). The spatial profiles of Vx and Vz are consistent with equation (4.22); they reveal cos (j^-x) and cos2 (^x) structures, respectively; in the vertical direction something similar takes place, up to the height where the system has been disturbed; and their maximum relative amplitudes are about 3 x 1 0 - 3 and 4 x 10~ 6 . On the other hand, the above scenario drastically changes when the driving frequency is reduced below the cut-off since perturbations become vertically evanescent. Figures 11.8c and d present the spatial profiles of Vx and Vz corresponding to CJJ = IT VA/L < OJC. Owing to the dispersive nature of the system in the linear regime, there are only high u, high kz propagating waves associated with Vx. Obviously, these modes possess frequencies above the cut-off and would not exist if the driver would be acting from t = - c o . In our numerical simulation, however, it starts acting at t = 0, so its spectrum of frequencies is not just the Fourier transform of sin(wd*), but that of H(t) sin(u)dt); see Eq. (11.18). The Fourier transform in time of this function contains a term inversely proportional to u^ — w2, so frequencies other than the driving one contribute to the temporal evolution of disturbances above the photosphere. The normal velocity component is next collected above the photosphere (at the point x = 0, z — 2L) and shown in Fig. 11.9a. Note that perturbations with the driving frequency are evanescent, so the amplitude of Vx is much smaller than that of the driver. The power spectrum (Fig. 11.9b) shows that there is no power below the cut-off frequency uc = ^ VA/L

Driven MHD waves in the solar corona

205

Fig. 11.8 Spatial distribution of (a) 14 and (b) Vz at t = § L/VA for a driver given by Eq. (11.18) with n = 3, V0 = 0.003 VA, and ud = ATV VA/L. This driving frequency lies well above the cut-off frequency UJC = Vkfe. (c) and (d) same as (a) and (b) but with Ud = ft VA/L, below the cut-off frequency. Note that perturbations in Vx and Vz are now purely evanescent; moreover, Vx reveals some high frequency, short wavelength waves propagating into the corona. and that nonlinearly driven modes appear at u = 3w
Numerical experiments

206

0.0004 0.0002 \

0.0000 -0.0002 -0.0004

\\ A A ft l\

h 1 / M In/lA W l Ml " V v : ' 4

6

B

10

12

tvyi 600

2" soo

i3ud

3 400

A

•

•

fe 300 t 200 3

S. ' 0 0

0 10

12

•

" i *

I

»0

.A 75

..J 20

25

Fig. 11.9 (a) Time evolution of Vx at a; = 0, z = 2 L for n = 3 and a driving amplitude Vb = 0.003 VA and frequency wd = 7r VA/L; see Eq. (11.18). (b) Spectrum corresponding to 14 in (a), showing the existence of nonlinearly driven modes above the cut-off frequency wc = 4p VA/L. Power around u; = 5 VA/L comes from the Fourier transform of the function H(t)sm(udt); see text, (c) Same as (a) with n = 5. Velocity oscillations seem to be of higher frequency than those in (a), (d) Same as (b); the cut-off frequency is now uic = ^ VA/L, so modes below this frequency become evanescent (compare (b) and (d)). waves now. In this figure some power at Ud (coming from linearly driven evanescent modes whose amplitude has not yet gone to zero at z — 2 L) and 7ujd (coming from nonlinear driving) can be also observed. Fig. 11.10, made for the larger amplitude Vo = 0.03 VA, allows us to check the effect of nonlinearities on the time signature and the corresponding power spectrum. In this case the amplitude of nonlinearly driven modes above the cut-off frequency is enhanced in comparison to the case of lower amplitude driver; compare Figs. 11.10b and Fig. 11.9b. Once the main nonlinear effects of the fast mode have been investigated, it is of interest to consider the influence of the detection point height above the photosphere. The horizontal velocity component at z = 2 L and z = 4 L are shown in Figs. 11.10a and c, respectively. Their power spectra

207

Driven MHD waves in the solar corona

0

5

10

15

20

25

Fig. 11.10 (a) Temporal evolution of Vx at x = 0, z = 2 L for a driver with Wd = ft VA/L, VO — 0.03 VA, and n = 3. (b) Power spectrum of Vx in (a); because of the larger driving amplitude, nonlinear effects are more important now (compare with Fig. 11.9b). (c) Same as (a) at x = 0, z — 4 L. (d) Power spectrum of Vx in (c). (Figs. 11.10b and d) look very similar and show the now familiar peaks at 3, 5, and 7 times the driving frequency, together with the peak coming from the power generated just above the cut-off frequency in the linear regime {LJL/VA — 5). This peak maintains the same power in the two spectra, but the other three appear to contain more power in the lower detection point. Nevertheless, the two signals have been recorded from t = 0 to WA/L = 12, so the one in Fig. 11.10a contains 10 time units of useful data, while the one in Fig. 11.10c contains only 8 time units of useful data. This difference causes the decrease in the power of the peaks at 3uid, 5cJd, and 7u>d, which can be proved by considering the time span 0 < WA/L < 10 at the low detection point before performing the spectral analysis. Then, the heights of peaks in the power spectrum become similar to those in Fig. ll.lOd, which allows us to conclude that in the present circumstances nonlinearities show no dependence with height. In this part of the book we have made reference to the normal veloc-

Numerical

208

experiments

ity component only. The density and vertical magnetic field component change very little from their equilibrium values during the computations; the horizontal magnetic field component remains almost equal to —Vx, as predicted by Eq. (11.19); and the parallel velocity component is nonlinearly driven by the normal component and its power spectrum reveals the existence of power at even multiples of the driving frequency. The effects on Vz, which always remains small compared to Vx, are more evident as the driving amplitude is increased to go from the linear to the nonlinear regime.

11.4.4

Summary

Using a simple magnetically inactive solar coronal hole model, namely, a two-dimensional Cartesian box with uniform vertical magnetic field, constant mass density, no gravity, and no pressure effects, we have investigated nonlinear fast magnetosonic waves which are excited at the foot-points of magnetic field lines by a periodic driver with a specific frequency and horizontal distribution. The slow and Alfven modes have been eliminated by considering the low plasma-/3 limit and by taking Vy — By = 0, d/dy = 0, respectively. Our model is complementary to recent numerical simulations (Murawski, Goossens 1994a, Murawski et al. 1996, Nakariakov, Roberts, Murawski 1997), which correspond to wave propagation along mass density enhancements in the presence of an inhomogeneous background flow, and we have benefited from these studies to gain an improved physical insight into the process of nonlinear wave propagation in straight magnetic field lines. In particular, we have learned that time signatures of the normal and parallel velocity components differ from those in the linear case, revealing that nonlinear effects give rise to new Fourier modes in these signatures. The numerical solution of the full system of MHD equations with a uniform (in x) driver, has allowed us to perform a direct comparison with the results coming from the model equations and an excellent agreement has been found. Finally, the photospheric structure of the driver has been included in our studies. Its main influence is to introduce a cut-off frequency below which modes become evanescent. This feature is a linear one that, in conjunction with the nonlinear picture in the case of a uniform driver, results in nonlinearly driven modes becoming evanescent when their frequencies lie below the cut-off.

Solar wind interaction

11.5

with Venus

209

Solar wind interaction with Venus

The planet Venus has been the subject of intense investigation since Mariner 2 flew by the planet in the Fall 1962. Observations of Venus by orbital missions have led to a significant improvement of our knowledge about the upper atmosphere and ionosphere of Venus and their interaction with the solar wind. Since the internal magnetic field of Venus is negligibly small or even nonexistent, the solar wind interaction with Venus differs from their terrestrial counterparts. This lack of magnetic field allows the solar wind to make direct contact with the ionosphere of the planet. Pioneer Venus Orbiter (PVO) as well as other spacecraft observations revealed that the solar wind interaction with Venus leads to a highly structured plasma (e. g., Phillips, McComas 1991). As a result of supersonic and superalfvenic solar wind flow, a bow shock forms upstream of the planet. The shock serves to slow, heat, and also assists in deflecting the solar wind. The shock which for average solar wind conditions is a standing fast magnetosonic wave departs itself from the obstacle so that the plasma that crossed the shock can flow around the planet. The size of the bow shock depends on the solar wind Mach numbers, solar wind dynamic pressure, as well as on the shape and the size of the ionosphere (Zhang et al. 1990). Apparent asymmetries in the shock shape result from the oblique to the solar wind flow interplanetary magnetic field (IMF) (Khurana, Kivelson 1994). Downstream of the bow shock, there exists a sharp gradient in the electron density known as ionopause. This is a region which separates the shocked and magnetized solar wind plasma from the thermal ionospheric plasma. The ionopause forms at the altitude above the surface of Venus where the ionospheric gas pressure is approximately balanced by the incident pressure in the overlaying magnetic barrier. The ionopause was observed to be typically located at about 300 km in the subsolar region and about 1000 km near the terminator (Phillips, McComas 1991). It is generally accepted that the height of the terminator ionopause affects the transport of ionospheric plasma to the nightside. On occasion when the solar wind dynamic pressure is high enough to substantially lower the terminator ionopause altitude, the nightside ionosphere observed by PVO is found to be highly depleted (Luhmann et al. 1987, Mahajan et al. 1989). This phenomenon is called the disappearing ionosphere. The region between the bow shock and the ionopause is referred to as the magnetosheath (Phillips, McComas 1991). The magnetosheath by itself

210

Numerical

experiments

contains a region (close to the ionopause) of enhanced magnetic pressure referred to as the magnetic barrier (Zhang et al. 1991). It is well known that Venus has a dayside exosphere which is dominated by oxygen at altitudes above 400 km from the planetary surface. The ionosphere is a partially-ionized component of exosphere above about 140 km from the surface of Venus. This region contains electrons and various ion species such as 0+, H+, Of, CO+, and others. The ionosphere is approximately in photochemical equilibrium below an altitude of about 200 km at Venus for all ions. Above 200 km, 0+ becomes the major ion in this region. The principal ionization source on the dayside is provided by solar photoionization of therrnospheric gases like O by solar extreme ultraviolet (EUV) radiation, although other ionization processes such as impact ionization and charge exchange may also contribute in a major way. On the nightside, solar photoionization does not contribute directly to the ionization, and the maintenance of the nightside ionosphere requires ion transport from the dayside through the terminator. The nightward ion flow is driven primarily by the large pressure gradient at the terminator. The ion flow generally increases with solar zenith angle (SZA), reaching values larger than 7 km/s downstream of the terminator (Brace, Hartle, Theis 1995). Ion-neutral chemical reactions and electron-ion charge exchange are both important processes in the lower ionosphere.

igtoFig. 11.11 Pressure distribution around Venus. The equatorial and meridian planes are horizontaland and vertical, respectively.

Solar wind interaction

with Venus

211

The observations of the nightside ionosphere provided an evidence that the ionospheric plasma is highly structured and dynamic (e. g., Brace, Kliore 1991), often exhibiting large-scale structures which are called tail rays. The ionosphere has a tendency to form a central tail ray, often with rays on either side, to the north and south. The rays have dimensions of the order of 1 - 3 x 103 km, decreasing in width at higher altitudes (Brace et al. 1987). Although the downstream extent of these structures is not measured since spacecraft orbits crossed them almost horizontally, it is supposed that they must extend tail ward at least few thousand kilometers downstream. In the nightside ionosphere, there are also regions of mass density depletions referred to as ionospheric holes (Brace et al. 1982). The density in these holes is lower than in the surrounding ionosphere by up to two orders of magnitude. The plasma in the holes differs from that found in their surrounding; H+ becomes a major ion in the holes, while 0+ is the major ion outside. These holes are associated with a strong magnetic field which points tailward (Marubashi et al. 1985). Most recent numerical simulations of the three-dimensional interaction between the solar wind and Venus largely improved our understanding of the large scale physical processes (Murawski, Steinolfson 1996a,b, Tanaka, Murawski 1997, Tanaka 1998a, 2000b). In particular, Murawski and Steinolfson (1996a) included mass loading due to photoionization of the oxygen atoms and show that the solar wind is decelerated by the mass loading and the bow shock is pushed farther outward from the planet. However, this model was developed for the case when the IMF is parallel to the solar wind flow, simplifying the geometry to two dimensions. This model was extended to three dimensions by Murawski and Steinolfson (1996b) and the case of the IMF perpendicular to the solar wind flow was considered. In another model, solar wind interaction with the ionosphere of Venus was numerically simulated in the framework of two-component, three-dimensional MHD model by Tanaka and Murawski (1997). This model is briefly described here. The effect of decreased ionospheric pressure which occurs under the condition of high speed solar wind or low solar extreme ultraviolet (EUV) flux, was discussed by Tanaka (1998b, 2000b). The results of numerical simulations showed that the IMF penetrates from the magnetosheath to the dayside ionosphere so as to increase the ionospheric total pressure. It is a purpose of this subsection to demonstrate that the basic features of the solar wind interaction with the ionosphere of Venus can be repro-

Numerical

212

experiments

duced by applying a two-component MHD model which was developed by Tanaka and Murawski (1997) and Tanaka (1998b, 2000b). This subsection is organized as follows. Numerical model of the dynamics of the solar wind and ionospheric plasma is reviewed in Sec. 11.5.1. Numerically obtained results and discussion are presented in the following section. This subsection closes with some concluding remarks.

11.5.1

Numerical

model

We assume that the neutral atmosphere of Venus consists of the oxygen atoms and of the carbon dioxide molecules which both are stratified gravitationally. Their number densities at the lower boundary of the atmosphere are 1010 1/cm3 and 5-10 10 1/cm3, respectively. The peak number densities occur at altitude 140 km above the planetary surface, in agreement with the PVO observations (Cravens et al. 1981). The ionosphere is approximately in photochemical equilibrium at lower altitudes. 0+ ions are produced primarily by the solar EUV incident on the neutral atmosphere, O + hu -¥ 0+, and by charge exchange with CO^ ions, COt +0^0++ C02. These chemical reactions occur with the production rates qi = 1 0 - 1 0 l/(cm 3 s) and q2 = 10~ 10 l/(cm 3 s), respectively. The density of C02 ions is calculated from the photo-chemical equilibrium. 0+ ions experience some losses during their charge exchange with molecules of the carbon dioxide, viz., 0+ + C02 -> CO + 0%. The loss rate for 0+ ions is Lx = 9.4 • 1 0 - 1 0 l/(cm 3 s). C0% ions are produced by the photoionization of the carbon dioxide molecules, C02 + hv -» CO%, and they experience charge losses during a chemical reaction with the oxygen atoms, viz., C02 + O -> 0% + CO. The loss rate for CO% ions is L2 = 1.64 • 1 0 - 1 0 l/(cm 3 s). We assume that the solar wind plasma consists of H+ ions which flow with the same velocity as 0+ ions. The set of equations used for a description of the solar wind interaction with Venus is that of two-component, ideal MHD that includes mass production and loss terms in the mass continuity equation, and aeronomical collision and gravity terms in the momentum equation. We solve the following set of MHD equations as an initial value problem

u, t + F i S + G,„ + H

z

= S.

(11.20)

Solar wind interaction with Venus

213

Here, the state vector of nine dependent variables is u = (g,mx,my,mz,Bx,By,Bz,E,Q2)T

(H-21)

and F, G, and H are flux functions in the x, y, and z directions (Tanaka 1998b, 2000b), respectively. The source term S depends on ion production due to photoionization and ion-neutral chemistry, qi, q2, as well as on losses due to ion-neutral reactions, Li, L2 (H+ ion-electron recombination is neglected in this model), viz., s

= (qi + 92 - L\ - L2, -urn. - gg, 0,0,0,

- — • (i/m + Qg) + -^-(qi

+ q2) ~ - ^ r ( £ i + L2),q2 - L2)T.

(11.22)

In the above formulae, g is the total ion density, g = g% + g2, with Qi and g2 corresponding to H+ ions and 0+ ions, respectively. The symbol m = QV — (rnx,my,Tnz) denotes the momentum. B is the magnetic field. E is the total energy density, v is the ion-neutral drag collision frequency. The ratio of specific heats is 7 = 5/3. Tq — 103 K is a production temperature of photoions and TL is a loss temperature due to a chemical reaction of the 0+ ions with the carbon dioxide. The other terms in the expressions above are self-explanatory. We assume that the magnetic field is perpendicular to the solar wind flow, while the IMF is typically oriented about 42° from the Sun-Venus line in the proper sense for an Archimedean spiral (Phillips, Luhmann, Russell 1986). As the perpendicular magnetic field case is simpler than the oblique field case, the present simulations will provide an insight into the more complex case. Consequently, the perpendicular magnetic field case seems to be motivated. Equation (11.20) is solved in all three spatial dimensions of a spherical r, 9, coordinate system by adopting a finite-volume method which uses a TVD scheme of Sec. 6.7.3 which was already successfully applied for a single-component plasma (Tanaka 1992, 1993b, 1994). For Eq. (11.20), the size of the Jacobian matrix increases to 9 x 9 from 8 x 8 for onecomponent MHD equations (Tanaka 1998b). The eigen-value problem for this Jacobian consists of two Alfven, two fast, two slow, and two entropy waves. Consequently, there is one more entropy wave in comparison to the eigen-waves of the Jacobian of the one-component MHD equations. Details

Numerical

214

experiments

of the present approach can be found elsewhere (Tanaka and Murawski 1997, Tanaka 1998b). The inner and outer boundaries of the simulation region are set at about 1 Rp and 10 Rp, respectively. Here Rp — 6053km + 140 km is the planetary radius. While the inflow boundary conditions are maintained on the day side of the outer boundary, the zero-gradient boundary conditions are adopted on the downstream side. Near the inner boundary, the ion-neutral collision and ion chemical processes become dominant. Therefore, an ion chemical equilibrium and zero plasma velocity conditions are adopted at the inner boundary. The ion temperature is fixed and held constant at the inner boundary throughout the simulation process. The simulation code used a 88 x 80 x 86 grid points along r x 6 x <j> directions. This grid provides the angular grid spacings Ad — 4.5° and A ~ 4°. The radial grid was chosen nonuniform with a finest grid of 0.00025 Rp at the inner boundary of the simulation region. The coarsest grid of 0.33 Rp was set at the outer boundary. A typical computation begins with the introduction of the desired solar wind values in the dayside within the numerical box. The numerical solution then continues until the interaction process achieves an approximate steady state. 11.5.2

Numerical

results

and

discussion

We report here only some of the results from our simulations. More details can be found in Tanaka and Murawski (1997) and Tanaka (1998b). We present all numerical results for the following solar wind parameters: proton density ne = 14 c m - 3 , temperature T = 105 °K, sound speed 61 km/s, solar wind speed 311 km/s which gives sonic Mach number 5.1, the plasma /? = 0.6, and the magnetic field strength 15 nT. These parameters correspond to the maximum of solar activity (Phillips, Luhmann, Russell 1984). Fig. 11.12 shows the pressure profiles along the Sun-Venus line. In the upstream solar wind, kinetic pressure gV2 dominates gas pressure p and magnetic pressure B2/(2(x). At the bow shock, kinetic energy of the solar wind is converted into thermal energy. As a consequence of that, the gas pressure dominates over the kinetic pressure downstream the bow shock. The distance between the bow shock and the planetary surface is about 0.45 Rp ~ 2700 km, where Rp ~ 6053 km is the radius of Venus (Fig. 11.12). With a distance closer to the planetary surface, the magnetic pressure ac-

Solar wind interaction

with Venus

215

Total '

'

RAM

' 2 Radial Distance

(Rp)

Fig. 11.12 The distributions of the gas pressure p, the magnetic pressure PB, and the dynamic pressure gV2 along the Sun-Venus line on the subsolar side. The horizontal axis shows the radial distance normalized to the planet radius Rp and the vertical axis shows relative pressure values. Note the bow shock at the distance 0.45 Rp, the ionopause at the altitude about 0.04 Rp, and the magnetic barrier which corresponds to the maximum of the magnetic pressure. commodates itself as a result of competitive ionospheric gas pressure, while at the same time gas pressure decreases. This behavior is a consequence of formation of the magnetic barrier which location corresponds to the maximum of the magnetic pressure (Fig. 11.12). The magnetic barrier is supported by the gas pressure of cold ionospheric plasma. This pressure is maintained by an ionization and ion chemical processes in the planetary upper atmosphere. At the bottom of the ionosphere, gas pressure is provided by the laying below neutral atmosphere through ion-neutral collisions. The ionopause occurs at the place where the impacting solar wind pressure is balanced by the ionospheric pressure. From Fig. 11.12 it is seen that the dynamic pressure is negligibly small downstream the bow shock as at the bow shock, the supersonic solar wind flow is diverted into a subsonic flow. Therefore, the ionopause is placed at the point where the gas pressure equals the magnetic pressure, at the distance about 0.04RP ~ 240 km from the planetary surface. The altitude at which the ionopause is located

Numerical experiments

216

is smallest at the nose, and it grows monotonically with increasing SZA, reaching the largest altitude at the terminator. The ionopause altitude is about 1 Rp at the terminator (Tanaka, Murawski 1997).

Fig. 11.13 Draping of magnetic field lines around Venus. Fig. 11.13 shows the global configuration of the the magnetic field lines and plasma density from the final configuration of the numerical simulations. A view is from the tailside. The solar wind flows in from the left-hand side towards the planet. Brown lines indicate magnetic field lines which pile up at the bow shock, and then slip over the ionosphere, forming magnetotail. Magnetic field lines, after being dragged through the polar regions are convected equatorward by field line tension and solar wind flow toward the antisolar direction. The geometry of the magnetic field on the nightside is related to the draping of the solar wind magnetic field over the obstacle on the dayside. Some of the draped magnetic field apparently sinks into the wake of the planet to create lobes-like structures with sunward and anti-sunward directed magnetic field. 11.5.3

Concluding

remarks

We considered the interaction of the solar wind with the ionosphere of Venus using numerical solutions of the two-component, three-dimensional MHD equations. For these solutions solar wind consists mainly of H+ ions, while a primary component of the ionosphere consists of 0+ ions. Loss effects

Ion-acoustic

waves and solitary

waves

217

due to the interaction of 0+ ions with molecules of the carbon dioxide are introduced. Such modeling has generally been successful in reproducing characteristics of the solar wind interaction with Venus. The main results are the following: The solar wind interaction with Venus leads to the formation of the bow shock and the ionosphere which consists of cold, low speed, weakly magnetized 0+ ions. The ionosphere exhibits a blunt conic shape, with a highly structured ionotail. With a growing distance from the planetary surface, the ionotail is flattened and that flattening is believed to be due to magnetic field tension forces. The present results can in principle be applied to any unmagnetized body that has an ionosphere. In particular, the results are expected to be quite relevant to Mars (e. g., Brecht, Ferrante 1991), several comets (e. g., Murawski et al. 1998), and a moon of Saturn - Titan (e. g., Keller, Cravens 1994).

11.6

Ion-acoustic waves and solitary waves

While studying ion-acoustic waves we should like to investigate the solutions of Eq. (4.76), for whilst Eq. (4.77), for strong magnetic fields, might appear to be more appropriate for the solar atmosphere, we note that Q is at most 0.13 (e. g., Brosius et al. 1992, Nitta et al. 1991, Webb et al. 1987) in coronal and 0.1 (Witt, Lotko 1983) in auroral plasmas. Indeed Gurnett et al. (1993) consider the ratio of ion gyrofrequency to plasma frequency to be so small that they effectively ignore the magnetic field and look at the KP or KdV solution of Eq. (4.76). But the effect of a magnetic field on the behaviour of ion-acoustic waves is known to be significant (Infeld 1985, Infeld, Frycz 1987), and whilst there have been some attempts at probing the mysteries of Eq. (4.76) (Laedke, Spatschek 1982, Infeld, Frycz 1991) we must content ourselves, at present, with gaining knowledge of magnetic field effects via the ZK equation, Eq. (4.77) (Murawski, Edwin 1992, Edwin, Murawski 1995). In a three-dimensional unmagnetized plasma, small-amplitude flat (or planar) solitary waves are stable, but the situation changes if an external magnetic field is applied to the plasma (Frycz, Infeld 1989, Infeld, Frycz 1987). Planar solitary waves, propagating parallel to the magnetic field, described by Equation (10), are only stable if they are disturbed by sufficiently short wavelength perturbations, that is those perturbations satisfy-

Numerical experiments

218

ing k2L2 > 5 (where ky is a wave number). Under certain conditions the planar waves break up into bell-shaped pulses (Frycz, Infeld 1989, Iwasaki, Toh, Kawahara 1990, Spatschek et al. 1991, Murawski, Edwin 1992). In this part of the book we investigate the effect of perturbing planar waves in directions perpendicular to their direction of propagation. To do so we employ a numerical code which utilizes a fourth-order Runge-Kutta method to represent the time derivative in the Zakharov-Kuznetsov equation. The spatial derivative terms were treated in a configuration space, transforming the wave function u(x,y,t) back and forth between configuration and Fourier spaces using a fast Fourier transform method (see Sec. 6.12.1 for a description of this method). Two-dimensional periodic domain: - 3 2 < x < 32, -2n/ky < y < 2n/ky was analyzed by 128 x 128 Fourier modes. The numerical results were verified by performing the standard tests, doubling the number of Fourier modes and halving the time-step until no significant changes appeared in u{x, y,t). Moreover, conservation laws for the Zakharov-Kuznetsov equation were used to check the numerical accuracy. The error was less than 1%. To study the evolution of flat solitary waves, the initial condition u(x,y,t

= 0) =uQsech(^j^-

j [1+ 0.3sin(fc„j/)]

(11.23)

was chosen. It represents a flat or planar solitary wave perturbed sinusoidally in the y-direction, perpendicular to the background-field direction, which is also its direction of propagation, the a;—direction. Infeld and Frycz (1987) showed that the development of this solitary wave depends upon the wave number of the initial perturbation. If the wave number is sufficiently large, k2L2 > 5, then the ion-acoustic solitary wave is stable. By comparison, a similar initial perturbation but with k2L2 = 0.1 will, after a while, develop into a cylindrical solitary wave. Fig. 11.14 shows such perturbed solitary wave at t = 57. This wave contains a new emerging structure which in a while (at t = 91) possesses a cylindrical symmetry. This is the scenario predicted by Frycz and Infeld (1989) and Spatschek et al. (1991). Numerical results lead to the following conclusions: A magnetic environment affects the stability of an ion-acoustic wave. A nonlinear planar ion-acoustic wave in a magnetic field-free region is stable, in a magnetic field it is stable only if perturbed by short wavelength perturbations. If perturbed by long wavelength perturbations even unstable flat waves can

Ion-acoustic waves and solitary waves

219

A

Fig. 11.14 Spatial profiles of a perturbed (with h\l? = 0.1) flat solitary wave of the Zakharov-Kuznetsov equation. This solitary wave propagates to the right with its initial amplitude UQ = 1.08: t = 57 (left panel); t = 91 (right panel). become stable (Prycz, Infeld 1989), as they are transformed into cylindrical solitary waves. Though we appear to have demonstrated this stability by giving only one example of a perturbation which produced cylindrical solitary waves, there are other examples in which stable features, cylindrical solitary waves, have been shown to emerge from unstable ones (Murawski, Edwin 1992, Iwasaki, Ton, Kawahara 1990). Though we know that ion-acoustic waves are not the simple waves described by the Korteweg-de Vries equation, but that they may be described, in a magnetic plasma, by the Zakharov-Kuznetsov equation, we do not yet have a complete picture of these waves. This section makes a contribution to the picture by demonstrating the theoretically predicted stability of a solitary wave propagating in the direction of the background magnetic field. Such a wave, if perturbed perpendicular to its propagation direction, it can be transformed into a cylindrical solitary wave. The resulting cylindrical solitary waves are more robust than their flat wave counterparts, preserving their cylindrical identities indefinitely in an ideal medium. As theoretical methods develop even further and more numerical and laboratory experiments are performed, we learn more about the features of these waves, putting us in a stronger position to try to explain some of the features observed in the auroral plasma and solar wind. And if the suggested measurements of Foukal and Hinata (1991) prove fruitful, perhaps ion-acoustic waves will be identifed with features in the solar corona too.

Chapter 12

Summary of the book

This monograph presents a few analytical and numerical methods for wave propagation in fluids. Although this presentation is far from complete the emphasis is on the methods which are the most effective and the best known for the author. The analytical methods have difnculties to deal with the full set of fluid equations. Instead, the analytical approach for solving the Cauchy problem for fluid equations is based on simplification of these equations to a model equation such as for instance the Burgers equation. This equation can be subsequently treated by various analytical methods. This simplification is usually accomplished for uni-directional flow and low but finite-amplitude perturbations. On the other hand, the numerical methods can treat both the model equations and the original set of the fluid equations such as the Euler equations or equations of magnetohydrodynamics. So, numerical simulations can be a useful (complementary to analysis and experiments) tool in studying fluids. As a consequence of numerical errors, numerically obtained results should be verified against analysis and experiments and even so have to be accepted with a doze of scepticisim. There are several conditions that numerical schemes should satisfy: accuracy and speed of numerical simulations, adequate representation of complex flows and steep profiles, without generation spurious oscillations as well as robustness. A computer code is described as being robust if it has the virtue of giving reliable results to a wide range of problems without needing to be retuned. Modern numerical schemes such as shock-capturing schemes described in this monograph satisfy these conditions. Existing numerical models demonstrate the feasibility of fluid simulations in obtaining at least qualitative and, to some extent, quantitative fea-

221

222

Summary of the book

tures in the fluid. With continued improvements in computational methods and computer resources, the usefulness and capability of the numerical approach should continue to improve. In particular, considering how useful the simulations have been, it is natural to propose replacing time consuming explicit methods by more efficient implicit methods as well as adopting modern techniques for spatial discretization such as shock-capturing schemes and application of adaptive mesh refinement methods. Another important problem is the adaptability of numerical schemes for parallel computations. Many schemes presented in this book are well suited for parallel computations and indeed such schemes were already developed.

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Index

acoustic cut-off frequency, 33 Adams method, 91 adaptive grid, 156 ADI scheme, 184 advection equation, 8, 81 air-pollutions, 194 Alfven speed, 28, 47, 199 Alfven wave, 28, 200 algebraic grid, 154 aliasing, 118 Ampere law, 15 amplification factor, 93 AMR-MHD scheme, 179 analysis, 1 anti-diffusion, 115

buoyancy (Brunt-Vaisala) frequency, 34 Burgers equation, 45, 50, 66, 188 capillary-gravity waves, 124 carbuncles, 144 causality, 103 Cauchy (initial-value) problem, 9 cell vertex scheme, 93 central difference, 80 CFL condition, 94 characteristics, 10, 127 Chebyshev, 119, 124 CLAWPACK, 191, 195 cold MHD equations, 23, 45 Cole-Hopf transformation, 68 collocation, 117, 119 compression, 10 conservation equation, 10, 114 conservative picture, 21 constraint transport, 174 contact discontinuity, 135, 137, 138 convection, 10, 34, 35 coordinate stretching, 47 coronal holes, 199 correlation, 38, 92 Crank-Nicolson scheme, 123 current density, 15 cut-off errors, 97 cut-off frequency, 33, 35, 204

backward difference, 80 Backlund transformation, 69 barely implicit scheme, 161, 184 Beam-Warming method, 84 Benjamin-Bona-Mahoney equation, 61 Benjamin-Ono equation, 61 bi-directional waves, 9, 61 black soliton, 76 Boltzmann's constant, 16 Boussinesq equation, 61 bow shock, 209 bright soliton, 75 buoyancy force, 196

235

236

cyclotron frequency, 15 cylinder, 52, 58 cylindrical KdV equation, 57 dark soliton, 78 Dawson integral, 39 Debye length, 16 Delaunay grid, 154, 155 density scale-height, 33 derivative expansion method, 58 diffusion equation, 7 diffusivity, 20 Dirac's delta, 41 direct method, 66 discretization error, 98 dispersion, 28, 32 dispersionless noise, 41 dispersive errors, 99 distribution function, 16 divergence cleaning, 173 divergence wave, 168 donor-cell, 87 downwind scheme, 88 Earth, 14 Earth's magnetosphere, 14 eigenvalue, 127, 133, 143, 170 eigenvector, 127, 133, 143, 170 electric conductivity, 15 electromagnetic waves, 25 electrostatic waves, 18, 25 elliptic equation, 7 elliptic grid, 154 energy equation, 21 enthalpy, 134 entropy, 134, 144 entropy fix, 144 error function, 121 Euler equations, 13, 137 Eulerian time derivative, 11 Eulerian picture, 21 Euler-Lagrange equation, 55 experiment, 1 explicit predictor, 162

Index Faraday law, 15 fast speed, 28, 198 fast wave, 29, 199 FCT method, 113, 200 finite-difference schemes, 81 finite-element, 121 finite-volume, 91 flux limiter, 116 Fourier method, 118 fractional step method, 141 frequency shift, 40, 43 Fromm's scheme, 87 front tracking methods, 111 Galerkin, 116 Gaussian spectrum, 39 Gel'fand-Levitan-Marchenko equation, 73 generalized density, 9 Gibbs errors, 99 Ginzburg-Landau equation, 60 Godunov method, 108 Godunov-splitting, 105 global error, 98 gravity, 33 gravity waves, 36 gyroradius, 16 Heaviside function, 202 Hermitian compact scheme, 85 HLL Riemann solver, 139 homogeneous driver, 202 hybrid approach, 16 hydrodynamic schemes, 163 hyperbolic system, 150 implicit pressure correction, 162 implicit scheme, 95, 158, 184 incompressible limit, 22 induction equation, 20 inertial length, 16 inhomogeneous driver, 204 instabilities, 96 internal gravity waves, 36

Index inverse scattering method, 71 inviscid Burgers equation, 45, 47, 49 ion-acoustic waves, 30, 217 ionopause, 209 ion-plasma frequency, 32 ion-sound speed, 26, 31 ionospheric holes, 211 isothermal atmosphere, 34 Jacobian, 139, 159, 170 jump condition, 136 Kadomtsev-Petviashvili (KP) equation, 58, 62 KdV-Burgers equation, 57 Kelvin-Helmholtz instabilities, 196 Korteweg-de Vries (KdV) equation, 52, 56, 69, 123 Lagrangian derivative, 12, 22 Lagrangian method, 55 Lagrangian picture, 22 Langmuir oscillations, 26 Laplace'a equation, 8 Lax criterion, 71 Lax-Predrichs scheme, 83 Lax-Wendroff scheme, 83 leap-frog method, 83 Leibovich-Pritchard-Roberts equation, 125 linearization, 27 Lorentz force, 21, 29 MacCormack scheme, 85 magnetic barrier, 210 magnetic pressure, 21 magnetic tension, 21 magnetic vector potential, 174 magnetohydromagnetic waves, 26 magnetosheath, 209 magnetosonic waves, 29 mass continuity equation, 11 mass density, 19 Maxwell's equations, 14

Maxwell-Boltzmann statistics, 31 mean free path, 15 measurements errors, 97 MHD equations, 21, 167 MHD waves, 26 microscopic description, 17 minmod function, 110 modeling errors, 107 modified KdV (mKdV) equation, 56 momentum density, 19 momentum equation, 21 monotonicity, 101 multigrid method, 90 multi-step methods, 90 MUSCL scheme, 109 Navier-Stokes equations, 12 Newtonian fluids, 12 non-conservative form, 168 nonlinear advection equation, 9 nonlinear effects, 189 nonlinear Schrodinger equation, 58, 74 number density, 17 numerical diffusion, 98 numerical dissipation, 98 numerical simulations, 2 Nyquist's critical wave vector, 118 Nystrom method, 91 Ohm's law, 15 operator splitting, 150 overshoots, 100 parabolic equation, 7 particle momenta, 19 p-characteristics, 129 periodic waves, 75 permeability, 15 permittivity, 15 phase speed error, 99 Pfaffian system, 69 piecewise linear interpolation, 109 piecewise parabolic interpolation, 111

237

238

Index

plasma, 2, 14 plasma dispersion function, 39 plasma frequency, 15, 25 plume, 196 Poisson equation, 7, 175 positivity, 101 potential, 19 power spectrum estimation, 189 predictor-corrector method, 114 pressure scale-height, 33 projection scheme, 109 Property U, 140 protons, 14 quasilinear form, 131 quasi-particles, 19 random field, 36, 191 random frequency, 39 random phase, 192 random waves, 36, 191 Rankine-Hugoniot, 136 rarefaction, 10, 135, 137 rarefaction fan, 135, 145 refinement criteria, 157 relaxation scheme, 141 resistivity, 20 reversibility, 104 Riemann invariants, 134 Riemann problem, 108, 129, 139, 173 Roe solver, 108, 139, Rossby waves, 63 round-off error, 98 Runge-Kutta methods, 89 saw-tooth oscillations, 202 scalar advection equation, 9 scalar potential, 175 Schwarzschild criterion, 34 seeding random field, 192 semi-implicit method, 95 semi-unstructured grid, 153 shock, 68, 76, 107, 135 singularities, 171

sinks, 13 slow wave, 29 Sod problem, 136 solar corona, 199 solar coronal loops, 29 solar wind, 209 solitary wave, 70, 74 soliton, 70, 72, 75, 78 Sommerfeld condition, 105 sonic point, 146 sound speed, 27 sound waves, 33, 36, 191 source term, 12, 104 specific heats ratio, 12 spherical KdV equation, 57 splitting error, 150 spurious mode, 100 statistical errors, 98 stiff source, 105 Strang splitting, 104 stratification, 33 stress, 12 structured grid, 152 Sturm-Liouville operator, 71 subcharacteristic condition, 141 subsonic flow, 161 Sun, 14 Sutherland's law, 13 tail rays, 211 tau approximation, 117 Taylor's expansion, 79 terminator, 209 thermal conduction, 13 thermal speed, 16 total energy density, 12, 21 total variation, 101 trajectory, 19, 134 trapezoidal method, 95 triple umbilic, 172 truncation error, 80 TVD, 101, 179 undershoots, 100

undetermined coefficients, 81 uni-directional wave, 9 unstructured grid, 152 upwind scheme, 86 vacuum, 15 variance, 39, 42 V-cycle, 90 Venus, 209 Veronoi grid, 155 viscosity, 12, 13, 50 Vlasov equation, 17 von Neumann stable, 93 wave amplification, 40 wave attenuation, 42 wave breaking, 66 wave envelope, 74 wave equation, 30, 37, 61 wavelet theory, 163 wave noise, 41 Wronskian determinants, 78 Young's modulus, 53 Zakharov equations, 62 Zakharov-Kuznetsov (ZK) equation, 58, 217