Fundamentals of Computational Fluid Dynamics

Fundamentals of Computational Fluid Dynamics Harvard Lomax and Thomas H. Pulliam NASA Ames Research Center David W. Zingg University of Toronto Institute for Aerospace Studies August 26, 1999

Contents 1 INTRODUCTION

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Problem Speci cation and Geometry Preparation . . . . . . 1.2.2 Selection of Governing Equations and Boundary Conditions 1.2.3 Selection of Gridding Strategy and Numerical Method . . . 1.2.4 Assessment and Interpretation of Results . . . . . . . . . . . 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 CONSERVATION LAWS AND THE MODEL EQUATIONS 2.1 Conservation Laws . . . . . . . . . . . . 2.2 The Navier-Stokes and Euler Equations . 2.3 The Linear Convection Equation . . . . 2.3.1 Dierential Form . . . . . . . . . 2.3.2 Solution in Wave Space . . . . . . 2.4 The Diusion Equation . . . . . . . . . . 2.4.1 Dierential Form . . . . . . . . . 2.4.2 Solution in Wave Space . . . . . . 2.5 Linear Hyperbolic Systems . . . . . . . . 2.6 Problems . . . . . . . . . . . . . . . . . .

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3 FINITE-DIFFERENCE APPROXIMATIONS 3.1 Meshes and Finite-Dierence Notation 3.2 Space Derivative Approximations . . . 3.3 Finite-Dierence Operators . . . . . . 3.3.1 Point Dierence Operators . . . 3.3.2 Matrix Dierence Operators . . 3.3.3 Periodic Matrices . . . . . . . . 3.3.4 Circulant Matrices . . . . . . . iii

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3.4 Constructing Dierencing Schemes of Any Order . . . . . 3.4.1 Taylor Tables . . . . . . . . . . . . . . . . . . . . 3.4.2 Generalization of Dierence Formulas . . . . . . . 3.4.3 Lagrange and Hermite Interpolation Polynomials 3.4.4 Practical Application of Pade Formulas . . . . . . 3.4.5 Other Higher-Order Schemes . . . . . . . . . . . . 3.5 Fourier Error Analysis . . . . . . . . . . . . . . . . . . . 3.5.1 Application to a Spatial Operator . . . . . . . . . 3.6 Dierence Operators at Boundaries . . . . . . . . . . . . 3.6.1 The Linear Convection Equation . . . . . . . . . 3.6.2 The Diusion Equation . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 THE SEMI-DISCRETE APPROACH

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5 FINITE-VOLUME METHODS

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4.1 Reduction of PDE's to ODE's . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Model ODE's . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Generic Matrix Form . . . . . . . . . . . . . . . . . . . . 4.2 Exact Solutions of Linear ODE's . . . . . . . . . . . . . . . . . . . . 4.2.1 Eigensystems of Semi-Discrete Linear Forms . . . . . . . . . . 4.2.2 Single ODE's of First- and Second-Order . . . . . . . . . . . . 4.2.3 Coupled First-Order ODE's . . . . . . . . . . . . . . . . . . . 4.2.4 General Solution of Coupled ODE's with Complete Eigensystems 4.3 Real Space and Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Eigenvalue Spectrums for Model ODE's . . . . . . . . . . . . . 4.3.3 Eigenvectors of the Model Equations . . . . . . . . . . . . . . 4.3.4 Solutions of the Model ODE's . . . . . . . . . . . . . . . . . . 4.4 The Representative Equation . . . . . . . . . . . . . . . . . . . . . . 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model Equations in Integral Form . . . . . . . . . . . . . . . . . . . 5.2.1 The Linear Convection Equation . . . . . . . . . . . . . . . 5.2.2 The Diusion Equation . . . . . . . . . . . . . . . . . . . . . 5.3 One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . 5.3.1 A Second-Order Approximation to the Convection Equation 5.3.2 A Fourth-Order Approximation to the Convection Equation 5.3.3 A Second-Order Approximation to the Diusion Equation . 5.4 A Two-Dimensional Example . . . . . . . . . . . . . . . . . . . . .

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52 52 53 54 54 55 57 59 61 61 62 63 65 67 68 72 73 73 74 74 75 77 78 80

5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 TIME-MARCHING METHODS FOR ODE'S

6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Converting Time-Marching Methods to O
E's . . . . . . . . . . . . 6.3 Solution of Linear O
E's With Constant Coecients . . . . . . . . 6.3.1 First- and Second-Order Dierence Equations . . . . . . . . 6.3.2 Special Cases of Coupled First-Order Equations . . . . . . . 6.4 Solution of the Representative O
E's . . . . . . . . . . . . . . . . 6.4.1 The Operational Form and its Solution . . . . . . . . . . . . 6.4.2 Examples of Solutions to Time-Marching O
E's . . . . . . . 6.5 The  ;  Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Establishing the Relation . . . . . . . . . . . . . . . . . . . . 6.5.2 The Principal -Root . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Spurious -Roots . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 One-Root Time-Marching Methods . . . . . . . . . . . . . . 6.6 Accuracy Measures of Time-Marching Methods . . . . . . . . . . . 6.6.1 Local and Global Error Measures . . . . . . . . . . . . . . . 6.6.2 Local Accuracy of the Transient Solution (er ; jj ; er! ) . . . 6.6.3 Local Accuracy of the Particular Solution (er ) . . . . . . . 6.6.4 Time Accuracy For Nonlinear Applications . . . . . . . . . . 6.6.5 Global Accuracy . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Linear Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 The General Formulation . . . . . . . . . . . . . . . . . . . . 6.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 Two-Step Linear Multistep Methods . . . . . . . . . . . . . 6.8 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . . . . 6.9 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Implementation of Implicit Methods . . . . . . . . . . . . . . . . . . 6.10.1 Application to Systems of Equations . . . . . . . . . . . . . 6.10.2 Application to Nonlinear Equations . . . . . . . . . . . . . . 6.10.3 Local Linearization for Scalar Equations . . . . . . . . . . . 6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations 6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 STABILITY OF LINEAR SYSTEMS 7.1 Dependence on the Eigensystem 7.2 Inherent Stability of ODE's . . 7.2.1 The Criterion . . . . . . 7.2.2 Complete Eigensystems .

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7.4 7.5 7.6

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7.8 7.9

7.2.3 Defective Eigensystems . . . . . . . . . . . . . . Numerical Stability of O
E 's . . . . . . . . . . . . . . 7.3.1 The Criterion . . . . . . . . . . . . . . . . . . . 7.3.2 Complete Eigensystems . . . . . . . . . . . . . . 7.3.3 Defective Eigensystems . . . . . . . . . . . . . . Time-Space Stability and Convergence of O
E's . . . . Numerical Stability Concepts in the Complex -Plane . 7.5.1 -Root Traces Relative to the Unit Circle . . . 7.5.2 Stability for Small
t . . . . . . . . . . . . . . . Numerical Stability Concepts in the Complex h Plane 7.6.1 Stability for Large h. . . . . . . . . . . . . . . . 7.6.2 Unconditional Stability, A-Stable Methods . . . 7.6.3 Stability Contours in the Complex h Plane. . . Fourier Stability Analysis . . . . . . . . . . . . . . . . 7.7.1 The Basic Procedure . . . . . . . . . . . . . . . 7.7.2 Some Examples . . . . . . . . . . . . . . . . . . 7.7.3 Relation to Circulant Matrices . . . . . . . . . . Consistency . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

8 CHOICE OF TIME-MARCHING METHODS

8.1 Stiness De nition for ODE's . . . . . . . . . . . . 8.1.1 Relation to -Eigenvalues . . . . . . . . . . 8.1.2 Driving and Parasitic Eigenvalues . . . . . . 8.1.3 Stiness Classi cations . . . . . . . . . . . . 8.2 Relation of Stiness to Space Mesh Size . . . . . . 8.3 Practical Considerations for Comparing Methods . 8.4 Comparing the Eciency of Explicit Methods . . . 8.4.1 Imposed Constraints . . . . . . . . . . . . . 8.4.2 An Example Involving Diusion . . . . . . . 8.4.3 An Example Involving Periodic Convection . 8.5 Coping With Stiness . . . . . . . . . . . . . . . . 8.5.1 Explicit Methods . . . . . . . . . . . . . . . 8.5.2 Implicit Methods . . . . . . . . . . . . . . . 8.5.3 A Perspective . . . . . . . . . . . . . . . . . 8.6 Steady Problems . . . . . . . . . . . . . . . . . . . 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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9 RELAXATION METHODS

9.1 Formulation of the Model Problem . . . . . . . . . . . . . . 9.1.1 Preconditioning the Basic Matrix . . . . . . . . . . . 9.1.2 The Model Equations . . . . . . . . . . . . . . . . . . 9.2 Classical Relaxation . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Delta Form of an Iterative Scheme . . . . . . . . 9.2.2 The Converged Solution, the Residual, and the Error 9.2.3 The Classical Methods . . . . . . . . . . . . . . . . . 9.3 The ODE Approach to Classical Relaxation . . . . . . . . . 9.3.1 The Ordinary Dierential Equation Formulation . . . 9.3.2 ODE Form of the Classical Methods . . . . . . . . . 9.4 Eigensystems of the Classical Methods . . . . . . . . . . . . 9.4.1 The Point-Jacobi System . . . . . . . . . . . . . . . . 9.4.2 The Gauss-Seidel System . . . . . . . . . . . . . . . . 9.4.3 The SOR System . . . . . . . . . . . . . . . . . . . . 9.5 Nonstationary Processes . . . . . . . . . . . . . . . . . . . . 9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 MULTIGRID

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11 NUMERICAL DISSIPATION

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10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.1.1 Eigenvector and Eigenvalue Identi cation with Space Frequencies191 10.1.2 Properties of the Iterative Method . . . . . . . . . . . . . . . 192 10.2 The Basic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.3 A Two-Grid Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.1 11.2 11.3 11.4

One-Sided First-Derivative Space Dierencing The Modi ed Partial Dierential Equation . . The Lax-Wendro Method . . . . . . . . . . . Upwind Schemes . . . . . . . . . . . . . . . . 11.4.1 Flux-Vector Splitting . . . . . . . . . . 11.4.2 Flux-Dierence Splitting . . . . . . . . 11.5 Arti cial Dissipation . . . . . . . . . . . . . . 11.6 Problems . . . . . . . . . . . . . . . . . . . . .

12 SPLIT AND FACTORED FORMS

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12.1 The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.2 Factoring Physical Representations | Time Splitting . . . . . . . . . 218 12.3 Factoring Space Matrix Operators in 2{D . . . . . . . . . . . . . . . . 220

12.4 12.5 12.6 12.7

12.3.1 Mesh Indexing Convention . . . . . . . . . . . 12.3.2 Data Bases and Space Vectors . . . . . . . . . 12.3.3 Data Base Permutations . . . . . . . . . . . . 12.3.4 Space Splitting and Factoring . . . . . . . . . Second-Order Factored Implicit Methods . . . . . . . Importance of Factored Forms in 2 and 3 Dimensions The Delta Form . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS

13.1 The Representative Equation for Circulant Operators . . . . . . . 13.2 Example Analysis of Circulant Systems . . . . . . . . . . . . . . . 13.2.1 Stability Comparisons of Time-Split Methods . . . . . . . 13.2.2 Analysis of a Second-Order Time-Split Method . . . . . . 13.3 The Representative Equation for Space-Split Operators . . . . . . 13.4 Example Analysis of 2-D Model Equations . . . . . . . . . . . . . 13.4.1 The Unfactored Implicit Euler Method . . . . . . . . . . . 13.4.2 The Factored Nondelta Form of the Implicit Euler Method 13.4.3 The Factored Delta Form of the Implicit Euler Method . . 13.4.4 The Factored Delta Form of the Trapezoidal Method . . . 13.5 Example Analysis of the 3-D Model Equation . . . . . . . . . . . 13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA 249 A.1 A.2 A.3 A.4 A.5

Notation . . . . . . . . . . De nitions . . . . . . . . . Algebra . . . . . . . . . . Eigensystems . . . . . . . Vector and Matrix Norms

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B SOME PROPERTIES OF TRIDIAGONAL MATRICES B.1 B.2 B.3 B.4

Standard Eigensystem for Simple Tridiagonals . . Generalized Eigensystem for Simple Tridiagonals . The Inverse of a Simple Tridiagonal . . . . . . . . Eigensystems of Circulant Matrices . . . . . . . . B.4.1 Standard Tridiagonals . . . . . . . . . . . B.4.2 General Circulant Systems . . . . . . . . . B.5 Special Cases Found From Symmetries . . . . . . B.6 Special Cases Involving Boundary Conditions . .

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C THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS265

Chapter 1 INTRODUCTION 1.1

Motivation

The material in this book originated from attempts to understand and systemize numerical solution techniques for the partial dierential equations governing the physics of uid ow. As time went on and these attempts began to crystallize, underlying constraints on the nature of the material began to form. The principal such constraint was the demand for uni cation. Was there one mathematical structure which could be used to describe the behavior and results of most numerical methods in common use in the eld of uid dynamics? Perhaps the answer is arguable, but the authors believe the answer is armative and present this book as justi cation for that belief. The mathematical structure is the theory of linear algebra and the attendant eigenanalysis of linear systems. The ultimate goal of the eld of computational uid dynamics (CFD) is to understand the physical events that occur in the ow of uids around and within designated objects. These events are related to the action and interaction of phenomena such as dissipation, diusion, convection, shock waves, slip surfaces, boundary layers, and turbulence. In the eld of aerodynamics, all of these phenomena are governed by the compressible Navier-Stokes equations. Many of the most important aspects of these relations are nonlinear and, as a consequence, often have no analytic solution. This, of course, motivates the numerical solution of the associated partial dierential equations. At the same time it would seem to invalidate the use of linear algebra for the classi cation of the numerical methods. Experience has shown that such is not the case. As we shall see in a later chapter, the use of numerical methods to solve partial dierential equations introduces an approximation that, in eect, can change the form of the basic partial dierential equations themselves. The new equations, which 1

CHAPTER 1. INTRODUCTION

2

are the ones actually being solved by the numerical process, are often referred to as the modi ed partial dierential equations. Since they are not precisely the same as the original equations, they can, and probably will, simulate the physical phenomena listed above in ways that are not exactly the same as an exact solution to the basic partial dierential equation. Mathematically, these dierences are usually referred to as truncation errors. However, the theory associated with the numerical analysis of

uid mechanics was developed predominantly by scientists deeply interested in the physics of uid ow and, as a consequence, these errors are often identi ed with a particular physical phenomenon on which they have a strong eect. Thus methods are said to have a lot of \arti cial viscosity" or said to be highly dispersive. This means that the errors caused by the numerical approximation result in a modi ed partial dierential equation having additional terms that can be identi ed with the physics of dissipation in the rst case and dispersion in the second. There is nothing wrong, of course, with identifying an error with a physical process, nor with deliberately directing an error to a speci c physical process, as long as the error remains in some engineering sense \small". It is safe to say, for example, that most numerical methods in practical use for solving the nondissipative Euler equations create a modi ed partial dierential equation that produces some form of dissipation. However, if used and interpreted properly, these methods give very useful information. Regardless of what the numerical errors are called, if their eects are not thoroughly understood and controlled, they can lead to serious diculties, producing answers that represent little, if any, physical reality. This motivates studying the concepts of stability, convergence, and consistency. On the other hand, even if the errors are kept small enough that they can be neglected (for engineering purposes), the resulting simulation can still be of little practical use if inecient or inappropriate algorithms are used. This motivates studying the concepts of stiness, factorization, and algorithm development in general. All of these concepts we hope to clarify in this book. 1.2

Background

The eld of computational uid dynamics has a broad range of applicability. Independent of the speci c application under study, the following sequence of steps generally must be followed in order to obtain a satisfactory solution.

1.2.1 Problem Speci cation and Geometry Preparation

The rst step involves the speci cation of the problem, including the geometry, ow conditions, and the requirements of the simulation. The geometry may result from

1.2. BACKGROUND

3

measurements of an existing con guration or may be associated with a design study. Alternatively, in a design context, no geometry need be supplied. Instead, a set of objectives and constraints must be speci ed. Flow conditions might include, for example, the Reynolds number and Mach number for the ow over an airfoil. The requirements of the simulation include issues such as the level of accuracy needed, the turnaround time required, and the solution parameters of interest. The rst two of these requirements are often in con ict and compromise is necessary. As an example of solution parameters of interest in computing the ow eld about an airfoil, one may be interested in i) the lift and pitching moment only, ii) the drag as well as the lift and pitching moment, or iii) the details of the ow at some speci c location.

1.2.2 Selection of Governing Equations and Boundary Conditions

Once the problem has been speci ed, an appropriate set of governing equations and boundary conditions must be selected. It is generally accepted that the phenomena of importance to the eld of continuum uid dynamics are governed by the conservation of mass, momentum, and energy. The partial dierential equations resulting from these conservation laws are referred to as the Navier-Stokes equations. However, in the interest of eciency, it is always prudent to consider solving simpli ed forms of the Navier-Stokes equations when the simpli cations retain the physics which are essential to the goals of the simulation. Possible simpli ed governing equations include the potential- ow equations, the Euler equations, and the thin-layer Navier-Stokes equations. These may be steady or unsteady and compressible or incompressible. Boundary types which may be encountered include solid walls, in ow and out ow boundaries, periodic boundaries, symmetry boundaries, etc. The boundary conditions which must be speci ed depend upon the governing equations. For example, at a solid wall, the Euler equations require ow tangency to be enforced, while the Navier-Stokes equations require the no-slip condition. If necessary, physical models must be chosen for processes which cannot be simulated within the speci ed constraints. Turbulence is an example of a physical process which is rarely simulated in a practical context (at the time of writing) and thus is often modelled. The success of a simulation depends greatly on the engineering insight involved in selecting the governing equations and physical models based on the problem speci cation.

1.2.3 Selection of Gridding Strategy and Numerical Method

Next a numerical method and a strategy for dividing the ow domain into cells, or elements, must be selected. We concern ourselves here only with numerical methods requiring such a tessellation of the domain, which is known as a grid, or mesh.

CHAPTER 1. INTRODUCTION

4

Many dierent gridding strategies exist, including structured, unstructured, hybrid, composite, and overlapping grids. Furthermore, the grid can be altered based on the solution in an approach known as solution-adaptive gridding. The numerical methods generally used in CFD can be classi ed as nite-dierence, nite-volume, nite-element, or spectral methods. The choices of a numerical method and a gridding strategy are strongly interdependent. For example, the use of nite-dierence methods is typically restricted to structured grids. Here again, the success of a simulation can depend on appropriate choices for the problem or class of problems of interest.

1.2.4 Assessment and Interpretation of Results

Finally, the results of the simulation must be assessed and interpreted. This step can require post-processing of the data, for example calculation of forces and moments, and can be aided by sophisticated ow visualization tools and error estimation techniques. It is critical that the magnitude of both numerical and physical-model errors be well understood. 1.3

Overview

It should be clear that successful simulation of uid ows can involve a wide range of issues from grid generation to turbulence modelling to the applicability of various simpli ed forms of the Navier-Stokes equations. Many of these issues are not addressed in this book. Some of them are presented in the books by Anderson, Tannehill, and Pletcher [1] and Hirsch [2]. Instead we focus on numerical methods, with emphasis on nite-dierence and nite-volume methods for the Euler and Navier-Stokes equations. Rather than presenting the details of the most advanced methods, which are still evolving, we present a foundation for developing, analyzing, and understanding such methods. Fortunately, to develop, analyze, and understand most numerical methods used to nd solutions for the complete compressible Navier-Stokes equations, we can make use of much simpler expressions, the so-called \model" equations. These model equations isolate certain aspects of the physics contained in the complete set of equations. Hence their numerical solution can illustrate the properties of a given numerical method when applied to a more complicated system of equations which governs similar physical phenomena. Although the model equations are extremely simple and easy to solve, they have been carefully selected to be representative, when used intelligently, of diculties and complexities that arise in realistic two- and three-dimensional uid

ow simulations. We believe that a thorough understanding of what happens when

1.4. NOTATION

5

numerical approximations are applied to the model equations is a major rst step in making con dent and competent use of numerical approximations to the Euler and Navier-Stokes equations. As a word of caution, however, it should be noted that, although we can learn a great deal by studying numerical methods as applied to the model equations and can use that information in the design and application of numerical methods to practical problems, there are many aspects of practical problems which can only be understood in the context of the complete physical systems. 1.4

Notation

The notation is generally explained as it is introduced. Bold type is reserved for real physical vectors, such as velocity. The vector symbol ~ is used for the vectors (or column matrices) which contain the values of the dependent variable at the nodes of a grid. Otherwise, the use of a vector consisting of a collection of scalars should be apparent from the context and is not identi ed by any special notation. For example, the variable u can denote a scalar Cartesian velocity component in the Euler and Navier-Stokes equations, a scalar quantity in the linear convection and diusion equations, and a vector consisting of a collection of scalars in our presentation of hyperbolic systems. Some of the abbreviations used throughout the text are listed and de ned below. PDE ODE O
E RHS P.S. S.S. k-D   ~ bc O()

Partial dierential equation Ordinary dierential equation Ordinary dierence equation Right-hand side Particular solution of an ODE or system of ODE's Fixed (time-invariant) steady-state solution k-dimensional space Boundary conditions, usually a vector A term of order (i.e., proportional to) 

6

CHAPTER 1. INTRODUCTION

Chapter 2 CONSERVATION LAWS AND THE MODEL EQUATIONS We start out by casting our equations in the most general form, the integral conservation-law form, which is useful in understanding the concepts involved in nite-volume schemes. The equations are then recast into divergence form, which is natural for nite-dierence schemes. The Euler and Navier-Stokes equations are brie y discussed in this Chapter. The main focus, though, will be on representative model equations, in particular, the convection and diusion equations. These equations contain many of the salient mathematical and physical features of the full Navier-Stokes equations. The concepts of convection and diusion are prevalent in our development of numerical methods for computational uid dynamics, and the recurring use of these model equations allows us to develop a consistent framework of analysis for consistency, accuracy, stability, and convergence. The model equations we study have two properties in common. They are linear partial dierential equations (PDE's) with coecients that are constant in both space and time, and they represent phenomena of importance to the analysis of certain aspects of uid dynamic problems.

2.1 Conservation Laws Conservation laws, such as the Euler and Navier-Stokes equations and our model equations, can be written in the following integral form: Z

V (t2 )

QdV ;

Z

V (t1 )

QdV +

Z t2 I

t1

S (t)

n:FdSdt =

Z t2 Z

t1

V (t)

PdV dt

(2.1)

In this equation, Q is a vector containing the set of variables which are conserved, e.g., mass, momentum, and energy, per unit volume. The equation is a statement of 7

8

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS

the conservation of these quantities in a nite region of space with volume V (t) and surface area S (t) over a nite interval of time t2 ; t1 . In two dimensions, the region of space, or cell, is an area A(t) bounded by a closed contour C (t). The vector n is a unit vector normal to the surface pointing outward, F is a set of vectors, or tensor, containing the ux of Q per unit area per unit time, and P is the rate of production of Q per unit volume per unit time. If all variables are continuous in time, then Eq. 2.1 can be rewritten as d Z QdV + I n:FdS = Z PdV (2.2) dt V (t) S (t) V (t) Those methods which make various numerical approximations of the integrals in Eqs. 2.1 and 2.2 and nd a solution for Q on that basis are referred to as nite-volume methods. Many of the advanced codes written for CFD applications are based on the nite-volume concept. On the other hand, a partial derivative form of a conservation law can also be derived. The divergence form of Eq. 2.2 is obtained by applying Gauss's theorem to the ux integral, leading to @Q + r:F = P (2.3) @t where r: is the well-known divergence operator given, in Cartesian coordinates, by ! @ @ @ (2.4) r:  i @x + j @y + k @z : and i; j, and k are unit vectors in the x; y, and z coordinate directions, respectively. Those methods which make various approximations of the derivatives in Eq. 2.3 and nd a solution for Q on that basis are referred to as nite-dierence methods.

2.2 The Navier-Stokes and Euler Equations The Navier-Stokes equations form a coupled system of nonlinear PDE's describing the conservation of mass, momentum and energy for a uid. For a Newtonian uid in one dimension, they can be written as @Q + @E = 0 (2.5) @t @x with 2 3 2 3  u 3 2 0 6 7 6 7 6 Q = 66 u 777 ; 4 5

e

E

6 6 = 666 4

7 6 7 6 7 2 u + p 77 ; 666 5 4

u(e + p)

4  @u 3 @x 4 @u 3 u @x

+  @T @x

7 7 7 7 7 5

(2.6)

2.2. THE NAVIER-STOKES AND EULER EQUATIONS

9

where  is the uid density, u is the velocity, e is the total energy per unit volume, p is the pressure, T is the temperature,  is the coecient of viscosity, and  is the thermal conductivity. The total energy e includes internal energy per unit volume  (where  is the internal energy per unit mass) and kinetic energy per unit volume u2=2. These equations must be supplemented by relations between  and  and the uid state as well as an equation of state, such as the ideal gas law. Details can be found in Anderson, Tannehill, and Pletcher [1] and Hirsch [2]. Note that the convective

uxes lead to rst derivatives in space, while the viscous and heat conduction terms involve second derivatives. This form of the equations is called conservation-law or conservative form. Non-conservative forms can be obtained by expanding derivatives of products using the product rule or by introducing dierent dependent variables, such as u and p. Although non-conservative forms of the equations are analytically the same as the above form, they can lead to quite dierent numerical solutions in terms of shock strength and shock speed, for example. Thus the conservative form is appropriate for solving ows with features such as shock waves. Many ows of engineering interest are steady (time-invariant), or at least may be treated as such. For such ows, we are often interested in the steady-state solution of the Navier-Stokes equations, with no interest in the transient portion of the solution. The steady solution to the one-dimensional Navier-Stokes equations must satisfy @E = 0 (2.7) @x If we neglect viscosity and heat conduction, the Euler equations are obtained. In two-dimensional Cartesian coordinates, these can be written as @Q + @E + @F = 0 (2.8) @t @x @y with 2 3 2 3 2 2 q1  u 3 v 3 6 7 6 7 6 u2 + p 7 6 uv 7 7 7 7 Q = 664 qq2 775 = 664 u 7; E = 6 6 7; F = 6 6 2 7 (2.9) 5 4 5 4 5 v uv v + p 3 q4 e u(e + p) v ( e + p) where u and v are the Cartesian velocity components. Later on we will make use of the following form of the Euler equations as well: @Q + A @Q + B @Q = 0 (2.10) @t @x @y @E and B = @F are known as the ux Jacobians. The ux vectors The matrices A = @Q @Q given above are written in terms of the primitive variables, , u, v, and p. In order

10

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS

to derive the ux Jacobian matrices, we must rst write the ux vectors E and F in terms of the conservative variables, q1 , q2, q3 , and q4 , as follows:

E

F

2 6 6 6 6 6 = 666 6 6 6 4

E1

2 3 6 6 7 6 7 6 7 6 7 6 7 6 7 7=6 6 7 6 7 6 7 7 6 5 6 6 4

3 7 7 7 2 2 q2 ;1 q3 7 3 ;

( ; 1)q4 + 2 q1 ; 2 q1 77 7 7 7 q3 q2 7 7 q1 7 7 7  q3 q2 q  5

; 1 2 q q

4q12 ; 2 q212 + 3q12

(2.11)

2 6 6 6 6 6 = 666 6 6 6 4

3 2 F1 7 66 7 6 7 6 7 F2 7 66 7 7 = 666 7 F3 77 66 7 6 5 6 4 F4

3 7 7 7 7 q3 q2 7 q1 7 7 7 7 2 2 ( ; 1)q4 + 3;2 qq31 ; ;2 1 qq21 77 7 7 5  q2 q  3 q q q

; 1 3

4q13 ; 2 2q12 + q132

(2.12)

E2 E3 E4

q2

q3

We have assumed that the pressure satis es p = ( ; 1)[e ; (u2 + v2)=2] from the ideal gas law, where is the ratio of speci c heats, cp=cv . From this it follows that the ux Jacobian of E can be written in terms of the conservative variables as

A=

2 6 6 6 6 6 @Ei = 666 @qj 66 6 6 6 4

3 7 7 q  q  7 a21 (3 ; ) q21 (1 ; ) q31 ; 1 777 7 7  q  q  q  q  7 2 3 3 2 ; q1 q1 0 77 q1 q1 7 q  7 5 a41 a42 a43

q21

0

1

0

where

a21

= ;2 1 qq3 1

!2

; 3 ;2 qq2 1

!2

0

(2.13)

2.2. THE NAVIER-STOKES AND EULER EQUATIONS 2 ( ; 1)4

+ qq3 1

!2

a42 = qq4 ; ;2 1 43 qq2 1 1 ! ! a43 = ;( ; 1) qq2 qq3 1 1 and in terms of the primitive variables as

!2

a41 =

q2 q1

!3

2

!

2 6 6 6 6 6 A = 666 6 6 6 4

where

0

1

q2 q1

!3 5;

+ qq3 1

!2 3 5

11

q4 q1

!

q2 q1

!

(2.14)

0

0

a21 (3 ; )u (1 ; )v ( ; 1)

;uv

v

u

0

a41

a42

a43

u

3 7 7 7 7 7 7 7 7 7 7 7 5

(2.15)

a21 = ;2 1 v2 ; 3 ;2 u2

a41 = ( ; 1)u(u2 + v2) ; ue 

a42 = e ; ;2 1 (3u2 + v2 )

a43 = (1 ; )uv (2.16) Derivation of the two forms of B = @F=@Q is similar. The eigenvalues of the ux Jacobian matrices are purely real. This is the de ning feature of hyperbolic systems of PDE's, which are further discussed in Section 2.5. The homogeneous property of the Euler equations is discussed in Appendix C. The Navier-Stokes equations include both convective and diusive uxes. This motivates the choice of our two scalar model equations associated with the physics of convection and diusion. Furthermore, aspects of convective phenomena associated with coupled systems of equations such as the Euler equations are important in developing numerical methods and boundary conditions. Thus we also study linear hyperbolic systems of PDE's.

12

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS

2.3 The Linear Convection Equation 2.3.1 Dierential Form

The simplest linear model for convection and wave propagation is the linear convection equation given by the following PDE: @u + a @u = 0 (2.17) @t @x Here u(x; t) is a scalar quantity propagating with speed a, a real constant which may be positive or negative. The manner in which the boundary conditions are speci ed separates the following two phenomena for which this equation is a model: (1) In one type, the scalar quantity u is given on one boundary, corresponding to a wave entering the domain through this \in ow" boundary. No boundary condition is speci ed at the opposite side, the \out ow" boundary. This is consistent in terms of the well-posedness of a 1st-order PDE. Hence the wave leaves the domain through the out ow boundary without distortion or re ection. This type of phenomenon is referred to, simply, as the convection problem. It represents most of the \usual" situations encountered in convecting systems. Note that the left-hand boundary is the in ow boundary when a is positive, while the right-hand boundary is the in ow boundary when a is negative. (2) In the other type, the ow being simulated is periodic. At any given time, what enters on one side of the domain must be the same as that which is leaving on the other. This is referred to as the biconvection problem. It is the simplest to study and serves to illustrate many of the basic properties of numerical methods applied to problems involving convection, without special consideration of boundaries. Hence, we pay a great deal of attention to it in the initial chapters. Now let us consider a situation in which the initial condition is given by u(x; 0) = u0(x), and the domain is in nite. It is easy to show by substitution that the exact solution to the linear convection equation is then u(x; t) = u0(x ; at) (2.18) The initial waveform propagates unaltered with speed jaj to the right if a is positive and to the left if a is negative. With periodic boundary conditions, the waveform travels through one boundary and reappears at the other boundary, eventually returning to its initial position. In this case, the process continues forever without any

2.3. THE LINEAR CONVECTION EQUATION

13

change in the shape of the solution. Preserving the shape of the initial condition u0(x) can be a dicult challenge for a numerical method.

2.3.2 Solution in Wave Space

We now examine the biconvection problem in more detail. Let the domain be given by 0  x  2. We restrict our attention to initial conditions in the form u(x; 0) = f (0)eix (2.19) where f (0) is a complex constant, and  is the wavenumber. In order to satisfy the periodic boundary conditions,  must be an integer. It is a measure of the number of wavelengths within the domain. With such an initial condition, the solution can be written as u(x; t) = f (t)eix (2.20) where the time dependence is contained in the complex function f (t). Substituting this solution into the linear convection equation, Eq. 2.17, we nd that f (t) satis es the following ordinary dierential equation (ODE) df = ;iaf (2.21) dt which has the solution f (t) = f (0)e;iat (2.22) Substituting f (t) into Eq. 2.20 gives the following solution u(x; t) = f (0)ei(x;at) = f (0)ei(x;!t) (2.23) where the frequency, !, the wavenumber, , and the phase speed, a, are related by ! = a (2.24) The relation between the frequency and the wavenumber is known as the dispersion relation. The linear relation given by Eq. 2.24 is characteristic of wave propagation in a nondispersive medium. This means that the phase speed is the same for all wavenumbers. As we shall see later, most numerical methods introduce some dispersion; that is, in a simulation, waves with dierent wavenumbers travel at dierent speeds. An arbitrary initial waveform can be produced by summing initial conditions of the form of Eq. 2.19. For M modes, one obtains

u(x; 0) =

M X m=1

fm (0)eimx

(2.25)

14

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS

where the wavenumbers are often ordered such that 1  2 


 M . Since the wave equation is linear, the solution is obtained by summing solutions of the form of Eq. 2.23, giving

u(x; t) =

M X

m=1

fm (0)eim(x;at)

(2.26)

Dispersion and dissipation resulting from a numerical approximation will cause the shape of the solution to change from that of the original waveform.

2.4 The Diusion Equation 2.4.1 Dierential Form

Diusive uxes are associated with molecular motion in a continuum uid. A simple linear model equation for a diusive process is

@u =  @ 2 u (2.27) @t @x2 where  is a positive real constant. For example, with u representing the temperature, this parabolic PDE governs the diusion of heat in one dimension. Boundary conditions can be periodic, Dirichlet (speci ed u), Neumann (speci ed @u=@x), or mixed Dirichlet/Neumann. In contrast to the linear convection equation, the diusion equation has a nontrivial steady-state solution, which is one that satis es the governing PDE with the partial derivative in time equal to zero. In the case of Eq. 2.27, the steady-state solution must satisfy @2u = 0 (2.28) @x2

Therefore, u must vary linearly with x at steady state such that the boundary conditions are satis ed. Other steady-state solutions are obtained if a source term g(x) is added to Eq. 2.27, as follows: "

#

@u =  @ 2 u ; g(x) @t @x2

(2.29)

giving a steady state-solution which satis es

@ 2 u ; g(x) = 0 @x2

(2.30)

2.4. THE DIFFUSION EQUATION

15

In two dimensions, the diusion equation becomes " # @u =  @ 2 u + @ 2 u ; g(x; y) (2.31) @t @x2 @y2 where g(x; y) is again a source term. The corresponding steady equation is @ 2 u + @ 2 u ; g(x; y) = 0 (2.32) @x2 @y2 While Eq. 2.31 is parabolic, Eq. 2.32 is elliptic. The latter is known as the Poisson equation for nonzero g, and as Laplace's equation for zero g.

2.4.2 Solution in Wave Space

We now consider a series solution to Eq. 2.27. Let the domain be given by 0  x   with boundary conditions u(0) = ua, u() = ub. It is clear that the steady-state solution is given by a linear function which satis es the boundary conditions, i.e., h(x) = ua + (ub ; ua)x=. Let the initial condition be

u(x; 0) =

M X m=1

fm(0) sin m x + h(x)

(2.33)

where  must be an integer in order to satisfy the boundary conditions. A solution of the form M X u(x; t) = fm(t) sin mx + h(x) (2.34) m=1

satis es the initial and boundary conditions. Substituting this form into Eq. 2.27 gives the following ODE for fm : dfm = ;2 f (2.35) m m dt and we nd fm(t) = fm(0)e;2m t (2.36) Substituting fm(t) into equation 2.34, we obtain

u(x; t) =

M X m=1

fm (0)e;2mt sin m x + h(x)

(2.37)

The steady-state solution (t ! 1) is simply h(x). Eq. 2.37 shows that high wavenumber components (large m) of the solution decay more rapidly than low wavenumber components, consistent with the physics of diusion.

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS

16

2.5 Linear Hyperbolic Systems

The Euler equations, Eq. 2.8, form a hyperbolic system of partial dierential equations. Other systems of equations governing convection and wave propagation phenomena, such as the Maxwell equations describing the propagation of electromagnetic waves, are also of hyperbolic type. Many aspects of numerical methods for such systems can be understood by studying a one-dimensional constant-coecient linear system of the form @u + A @u = 0 (2.38) @t @x where u = u(x; t) is a vector of length m and A is a real m  m matrix. For conservation laws, this equation can also be written in the form @u + @f = 0 (2.39) @t @x @f is the ux Jacobian matrix. The entries in the where f is the ux vector and A = @u

ux Jacobian are @fi aij = @u (2.40) j The ux Jacobian for the Euler equations is derived in Section 2.2. Such a system is hyperbolic if A is diagonalizable with real eigenvalues.1 Thus  = X ;1AX (2.41) where  is a diagonal matrix containing the eigenvalues of A, and X is the matrix of right eigenvectors. Premultiplying Eq. 2.38 by X ;1 , postmultiplying A by the product XX ;1, and noting that X and X ;1 are constants, we obtain

@X ;1u

 z }| { ; 1 @ X AX X ;1u

=0 (2.42) @t + @x With w = X ;1u, this can be rewritten as @w +  @w = 0 (2.43) @t @x When written in this manner, the equations have been decoupled into m scalar equations of the form @wi +  @wi = 0 (2.44) i @t @x 1

See Appendix A for a brief review of some basic relations and de nitions from linear algebra.

2.6. PROBLEMS

17

The elements of w are known as characteristic variables. Each characteristic variable satis es the linear convection equation with the speed given by the corresponding eigenvalue of A. Based on the above, we see that a hyperbolic system in the form of Eq. 2.38 has a solution given by the superposition of waves which can travel in either the positive or negative directions and at varying speeds. While the scalar linear convection equation is clearly an excellent model equation for hyperbolic systems, we must ensure that our numerical methods are appropriate for wave speeds of arbitrary sign and possibly widely varying magnitudes. The one-dimensional Euler equations can also be diagonalized, leading to three equations in the form of the linear convection equation, although they remain nonlinear, of course. The eigenvalues of the ux Jacobian matrix, orqwave speeds, are u; u + c, and u ; c, where u is the local uid velocity, and c = p= is the local speed of sound. The speed u is associated with convection of the uid, while u + c and u ; c are associated with sound waves. Therefore, in a supersonic ow, where juj > c, all of the wave speeds have the same sign. In a subsonic ow, where juj < c, wave speeds of both positive and negative sign are present, corresponding to the fact that sound waves can travel upstream in a subsonic ow. The signs of the eigenvalues of the matrix A are also important in determining suitable boundary conditions. The characteristic variables each satisfy the linear convection equation with the wave speed given by the corresponding eigenvalue. Therefore, the boundary conditions can be speci ed accordingly. That is, characteristic variables associated with positive eigenvalues can be speci ed at the left boundary, which corresponds to in ow for these variables. Characteristic variables associated with negative eigenvalues can be speci ed at the right boundary, which is the in ow boundary for these variables. While other boundary condition treatments are possible, they must be consistent with this approach.

2.6 Problems 1. Show that the 1-D Euler equations can be written in terms of the primitive variables R = [; u; p]T as follows: @R + M @R = 0 @t @x where 2 3 u  0 M = 64 0 u ;1 75 0 p u

18

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS Assume an ideal gas, p = ( ; 1)(e ; u2=2). 2. Find the eigenvalues and eigenvectors of the matrix M derived in question 1. 3. Derive the ux Jacobian matrix A = @E=@Q for the 1-D Euler equations resulting from the conservative variable formulation (Eq. 2.5). Find its eigenvalues and compare with those obtained in question 2. 4. Show that the two matrices M and A derived in questions 1 and 3, respectively, are related by a similarity transform. (Hint: make use of the matrix S = @Q=@R.) 5. Write the 2-D diusion equation, Eq. 2.31, in the form of Eq. 2.2. 6. Given the initial condition u(x; 0) = sin x de ned on 0  x  2, write it in the form of Eq. 2.25, that is, nd the necessary values of fm (0). (Hint: use M = 2 with 1 = 1 and 2 = ;1.) Next consider the same initial condition de ned only at x = 2j=4, j = 0; 1; 2; 3. Find the values of fm(0) required to reproduce the initial condition at these discrete points using M = 4 with m = m ; 1. 7. Plot the rst three basis functions used in constructing the exact solution to the diusion equation in Section 2.4.2. Next consider a solution with boundary conditions ua = ub = 0, and initial conditions from Eq. 2.33 with fm(0) = 1 for 1  m  3, fm (0) = 0 for m > 3. Plot the initial condition on the domain 0  x  . Plot the solution at t = 1 with  = 1. 8. Write the classical wave equation @ 2 u=@t2 = c2 @ 2 u=@x2 as a rst-order system, i.e., in the form @U + A @U = 0 @t @x where U = [@u=@x; @u=@t]T . Find the eigenvalues and eigenvectors of A. 9. The Cauchy-Riemann equations are formed from the coupling of the steady compressible continuity (conservation of mass) equation @u + @v = 0 @x @y and the vorticity de nition @v + @u = 0 ! = ; @x @y

2.6. PROBLEMS

19

where ! = 0 for irrotational ow. For isentropic and homenthalpic ow, the system is closed by the relation 

  ;1 1

; 1 2 2 = 1; 2 u +v ;1 Note that the variables have been nondimensionalized. Combining the two PDE's, we have @f (q) + @g(q) = 0 @x @y where       u ; u ; v q = v ; f = v ; g = ;u One approach to solving these equations is to add a time-dependent term and nd the steady solution of the following equation: @q + @f + @g = 0 @t @x @y (a) Find the ux Jacobians of f and g with respect to q. (b) Determine the eigenvalues of the ux Jacobians. (c) Determine the conditions (in terms of  and u) under which the system is hyperbolic, i.e., has real eigenvalues. (d) Are the above uxes homogeneous? (See Appendix C.)

20

CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS












 



 

  





 

 

 

 









 





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Chapter 4 THE SEMI-DISCRETE APPROACH One strategy for obtaining nite-dierence approximations to a PDE is to start by dierencing the space derivatives only, without approximating the time derivative. In the following chapters, we proceed with an analysis making considerable use of this concept, which we refer to as the semi-discrete approach. Dierencing the space derivatives converts the basic PDE into a set of coupled ODE's. In the most general notation, these ODE's would be expressed in the form d~u = F~ (~u; t) (4.1) dt which includes all manner of nonlinear and time-dependent possibilities. On occasion, we use this form, but the rest of this chapter is devoted to a more specialized matrix notation described below. Another strategy for constructing a nite-dierence approximation to a PDE is to approximate all the partial derivatives at once. This generally leads to a point dierence operator (see Section 3.3.1) which, in turn, can be used for the time advance of the solution at any given point in the mesh. As an example let us consider the model equation for diusion @u =  @ u @t @x Using three-point central-dierencing schemes for both the time and space derivatives, we nd 2 n n +un 3 ujn ; ujn; u ; 2 u j j; 5 4 j =  2h
x or 2h hu n ; 2u n + u n i ujn = ujn; +
(4.2) j j; x j 51 2

2

( +1)

(

1)

( ) +1

( )

( ) 1

2

( +1)

(

1)

2

( ) +1

( )

( ) 1

CHAPTER 4. THE SEMI-DISCRETE APPROACH

52

Clearly Eq. 4.2 is a dierence equation which can be used at the space point j to advance the value of u from the previous time levels n and n ; 1 to the level n + 1. It is a full discretization of the PDE. Note, however, that the spatial and temporal discretizations are separable. Thus, this method has an intermediate semi-discrete form and can be analyzed by the methods discussed in the next few chapters. Another possibility is to replace the value of ujn in the right hand side of Eq. 4.2 with the time average of u at that point, namely (ujn + ujn; )=2. This results in the formula ( )

( +1)

ujn

( +1)

2

0

un = ujn; + 2h 4ujn ; 2 @ j
x (

1)

2

( ) +1

( +1)

(

1)

1

3

+ ujn; A n 5 + uj ; 2 (

1)

( ) 1

(4.3)

which can be solved for u n and time advanced at the point j . In this case, the spatial and temporal discretizations are not separable, and no semi-discrete form exists. Equation 4.2 is sometimes called Richardson's method of overlapping steps and Eq. 4.3 is referred to as the DuFort-Frankel method. As we shall see later on, there are subtle points to be made about using these methods to nd a numerical solution to the diusion equation. There are a number of issues concerning the accuracy, stability, and convergence of Eqs. 4.2 and 4.3 which we cannot comment on until we develop a framework for such investigations. We introduce these methods here only to distinguish between methods in which the temporal and spatial terms are discretized separately and those for which no such separation is possible. For the time being, we shall separate the space dierence approximations from the time dierencing. In this approach, we reduce the governing PDE's to ODE's by discretizing the spatial terms and use the well-developed theory of ODE solutions to aid us in the development of an analysis of accuracy and stability. ( +1)

4.1 Reduction of PDE's to ODE's 4.1.1 The Model ODE's First let us consider the model PDE's for diusion and biconvection described in Chapter 2. In these simple cases, we can approximate the space derivatives with dierence operators and express the resulting ODE's with a matrix formulation. This is a simple and natural formulation when the ODE's are linear.

4.1. REDUCTION OF PDE'S TO ODE'S

53

Model ODE for Diusion For example, using the 3-point central-dierencing scheme to represent the second derivative in the scalar PDE governing diusion leads to the following ODE diusion model d~u =  B (1; ;2; 1)~u + (bc ~) (4.4) dt
x ~ ) vector. with Dirichlet boundary conditions folded into the (bc 2

Model ODE for Biconvection The term biconvection was introduced in Section 2.3. It is used for the scalar convection model when the boundary conditions are periodic. In this case, the 3-point central-dierencing approximation produces the ODE model given by

d~u = ; a B (;1; 0; 1)~u (4.5) dt 2
x p where the boundary condition vector is absent because the ow is periodic. Eqs. 4.4 and 4.5 are the model ODE's for diusion and biconvection of a scalar in one dimension. They are linear with coecient matrices which are independent of x and t.

4.1.2 The Generic Matrix Form

The generic matrix form of a semi-discrete approximation is expressed by the equation

d~u = A~u ; ~f (t) dt

(4.6)

Note that the elements in the matrix A depend upon both the PDE and the type of dierencing scheme chosen for the space terms. The vector ~f (t) is usually determined by the boundary conditions and possibly source terms. In general, even the Euler and Navier-Stokes equations can be expressed in the form of Eq. 4.6. In such cases the equations are nonlinear, that is, the elements of A depend on the solution ~u and are usually derived by nding the Jacobian of a ux vector. Although the equations are nonlinear, the linear analysis presented in this book leads to diagnostics that are surprisingly accurate when used to evaluate many aspects of numerical methods as they apply to the Euler and Navier-Stokes equations.

54

CHAPTER 4. THE SEMI-DISCRETE APPROACH

4.2 Exact Solutions of Linear ODE's

In order to advance Eq. 4.1 in time, the system of ODE's must be integrated using a time-marching method. In order to analyze time-marching methods, we will make use of exact solutions of coupled systems of ODE's, which exist under certain conditions. The ODE's represented by Eq. 4.1 are said to be linear if F is linearly dependent on u (i.e., if @F=@u = A where A is independent of u). As we have already pointed out, when the ODE's are linear they can be expressed in a matrix notation as Eq. 4.6 in which the coecient matrix, A, is independent of u. If A does depend explicitly on t, the general solution cannot be written; whereas, if A does not depend explicitly on t, the general solution to Eq. 4.6 can be written. This holds regardless of whether or not the forcing function, f~, depends explicitly on t. As we shall soon see, the exact solution of Eq. 4.6 can be written in terms of the eigenvalues and eigenvectors of A. This will lead us to a representative scalar equation for use in analyzing time-marching methods. These ideas are developed in the following sections.

4.2.1 Eigensystems of Semi-Discrete Linear Forms

Complete Systems An M  M matrix is represented by a complete eigensystem if it has a complete set

of linearly independent eigenvectors (see Appendix A) . An eigenvector, ~xm , and its corresponding eigenvalue, m, have the property that A~xm = m~xm or [A ; mI]~xm = 0 (4.7) The eigenvalues are the roots of the equation det[A ; I] = 0 We form the right-hand eigenvector matrix of a complete system by lling its columns with the eigenvectors ~xm: h i X = ~x ; ~x : : : ; ~xM The inverse is the left-hand eigenvector matrix, and together they have the property that 1

2

X ; AX =  1

where  is a diagonal matrix whose elements are the eigenvalues of A.

(4.8)

4.2. EXACT SOLUTIONS OF LINEAR ODE'S

55

Defective Systems If an M  M matrix does not have a complete set of linearly independent eigenvectors,

it cannot be transformed to a diagonal matrix of scalars, and it is said to be defective. It can, however, be transformed to a diagonal set of blocks, some of which may be scalars (see Appendix A). In general, there exists some S which transforms any matrix A such that S ; AS = J where 2J 3 66 J 77 66 77 . .. J =6 77 64 Jm 5 ... and 2 1 3 m 1. 66 7 . . .. 7 . m 7 Jmn = 66 7 . . . 1 5 ... 4 m n The matrix J is said to be in Jordan canonical form, and an eigenvalue with multiplicity n within a Jordan block is said to be a defective eigenvalue. Defective systems play a role in numerical stability analysis. 1

1

2

( )

4.2.2 Single ODE's of First- and Second-Order First-Order Equations

The simplest nonhomogeneous ODE of interest is given by the single, rst-order equation du = u + aet (4.9) dt where , a, and  are scalars, all of which can be complex numbers. The equation is linear because  does not depend on u, and has a general solution because  does not depend on t. It has a steady-state solution if the right-hand side is independent of t, i.e., if  = 0, and is homogeneous if the forcing function is zero, i.e., if a = 0. Although it is quite simple, the numerical analysis of Eq. 4.9 displays many of the fundamental properties and issues involved in the construction and study of most popular time-marching methods. This theme will be developed as we proceed.

CHAPTER 4. THE SEMI-DISCRETE APPROACH

56

The exact solution of Eq. 4.9 is, for  6= , t ae u(t) = c ; where c is a constant determined by the initial conditions. In terms of the initial value of u, it can be written 1

et +

1

t ; et u(t) = u(0)et + a e  ; 

The interesting question can arise: What happens to the solution of Eq. 4.9 when  = ? This is easily found by setting  =  + , solving, and then taking the limit as  ! 0. Using this limiting device, we nd that the solution to

du = u + aet dt

(4.10)

is given by

u(t) = [u(0) + at]et As we shall soon see, this solution is required for the analysis of defective systems.

Second-Order Equations The homogeneous form of a second-order equation is given by

d u + a du + a u = 0 (4.11) dt dt where a and a are complex constants. Now we introduce the dierential operator D such that D  dtd 2

1

2

1

0

0

and factor u(t) out of Eq. 4.11, giving

(D + a D + a ) u(t) = 0 2

1

0

The polynomial in D is referred to as a characteristic polynomial and designated P (D). Characteristic polynomials are fundamental to the analysis of both ODE's and O
E's, since the roots of these polynomials determine the solutions of the equations. For ODE's, we often label these roots in order of increasing modulus as  ,  ,


, m , 1

2

4.2. EXACT SOLUTIONS OF LINEAR ODE'S

57




, M . They are found by solving the equation P () = 0. In our simple example,

there would be two roots,  and  , determined from P () =  + a  + a = 0 (4.12) and the solution to Eq. 4.11 is given by u(t) = c e1 t + c e2 t (4.13) where c and c are constants determined from initial conditions. The proof of this is simple and is found by substituting Eq. 4.13 into Eq. 4.11. One nds the result c e1 t ( + a  + a ) + c e2 t ( + a  + a ) which is identically zero for all c , c , and t if and only if the 's satisfy Eq. 4.12. 1

2

2

1

0

1

1

2

2

1

2 1

1

1

1

0

2 2

2

1

2

0

2

4.2.3 Coupled First-Order ODE's A Complete System

A set of coupled, rst-order, homogeneous equations is given by u0 = a u + a u u0 = a u + a u (4.14) which can be written " # 0 a a ~u = A~u ; ~u = [u ; u ]T ; A = (aij ) = a a Consider the possibility that a solution is represented by u = c x e1 t + c x e2 t u = c x e1 t + c x e2 t (4.15) By substitution, these are indeed solutions to Eq. 4.14 if and only if " #" # " # " #" # " # a a x = x a a x x ; =  (4.16) a a x x a a x x Notice that a higher-order equation can be reduced to a coupled set of rst-order equations by introducing a new set of dependent variables. Thus, by setting u = u0 ; u = u ; we nd Eq. 4.11 can be written u0 = ;a u ; a u u0 = u (4.17) which is a subset of Eq. 4.14. 1

11

1

12

2

2

21

1

22

2

1

11

12

11

21

22

21

2

1

1

11

2

12

2

1

21

2

22

1

11

11

12

12

21

21

22

22

1

1

2

2

1

1 1

0

2

11

12

21

22

2

12 22

CHAPTER 4. THE SEMI-DISCRETE APPROACH

58

A Derogatory System Eq. 4.15 is still a solution to Eq. 4.14 if  =  = , provided two linearly independent vectors exist to satisfy Eq. 4.16 with A = . In this case 1

 01 1 0  0 = 0

2

 0 0 0 0  1 = 1

and

provide such a solution. This is the case where A has a complete set of eigenvectors and is not defective.

A Defective System If A is defective, then it can be represented by the Jordan canonical form

"

# " #" # u0 =  0 u u0 1  u 1

1

2

2

(4.18)

whose solution is not obvious. However, in this case, one can solve the top equation rst, giving u (t) = u (0)et . Then, substituting this result into the second equation, one nds du = u + u (0)et dt which is identical in form to Eq. 4.10 and has the solution 1

1

2

2

1

u (t) = [u (0) + u (0)t]et 2

2

1

From this the reader should be able to verify that

  u (t) = a + bt + ct et 2

3

is a solution to

2 03 2 32 3 u  64 u0 75 = 64 1  75 64 uu 75 u0 1  u 1

1

2

2

3

3

if

a = u (0) ; b = u (0) ; c = 21 u (0) The general solution to such defective systems is left as an exercise. 3

2

1

(4.19)

4.2. EXACT SOLUTIONS OF LINEAR ODE'S

59

4.2.4 General Solution of Coupled ODE's with Complete Eigensystems Let us consider a set of coupled, nonhomogeneous, linear, rst-order ODE's with constant coecients which might have been derived by space dierencing a set of PDE's. Represent them by the equation

d~u = A~u ; ~f (t) (4.20) dt Our assumption is that the M  M matrix A has a complete eigensystem and can be transformed by the left and right eigenvector matrices, X ; and X , to a diagonal matrix  having diagonal elements which are the eigenvalues of A, see Section 4.2.1. Now let us multiply Eq. 4.20 from the left by X ; and insert the identity combination XX ; = I between A and ~u. There results ~ X ; ddtu = X ; AX
X ; ~u ; X ; ~f (t) (4.21) Since A is independent of both ~u and t, the elements in X ; and X are also independent of both ~u and t, and Eq. 4.21 can be modi ed to 1

1

1

1

1

1

1

1

1

d X ; ~u = X ; ~u ; X ; ~f (t) dt Finally, by introducing the new variables w~ and ~g such that 1

1

1

w~ = X ; ~u ; ~g(t) = X ; ~f (t) 1

1

(4.22)

we reduce Eq. 4.20 to a new algebraic form

dw~ = w~ ; ~g(t) (4.23) dt It is important at this point to review the results of the previous paragraph. Notice that Eqs. 4.20 and 4.23 are expressing exactly the same equality. The only dierence between them was brought about by algebraic manipulations which regrouped the variables. However, this regrouping is crucial for the solution process because Eqs. In the following, we exclude defective systems, not because they cannot be analyzed (the example at the conclusion of the previous section proves otherwise), but because they are only of limited interest in the general development of our theory. 1

60

CHAPTER 4. THE SEMI-DISCRETE APPROACH

4.23 are no longer coupled. They can be written line by line as a set of independent, single, rst-order equations, thus

w0 =  w ; g (t) ... wm0 = mwm ; gm (t) ... wM0 = M wM ; gM (t) 1

1

1

1

(4.24)

For any given set of gm (t) each of these equations can be solved separately and then recoupled, using the inverse of the relations given in Eqs. 4.22: ~u(t) = X w~ (t) =

M X

m=1

wm(t)~xm

(4.25)

where ~xm is the m'th column of X , i.e., the eigenvector corresponding to m . We next focus on the very important subset of Eq. 4.20 when neither A nor ~f has any explicit dependence on t. In such a case, the gm in Eqs. 4.23 and 4.24 are also time invariant and the solution to any line in Eq. 4.24 is wm (t) = cm em t + 1 gm m where the cm are constants that depend on the initial conditions. Transforming back to the u-system gives ~u(t) = X w~ (t) =

M X

m=1 M X

wm(t)~xm

M 1 X gm~xm  m m m M X = cm em t ~xm + X ; X ; ~f

=

=1

cm em t ~xm +

=1

1

=

m=1 M X m=1

|

1

cm em t ~xm + A; ~f 1

{z

Transient

} |

{z

}

Steady-state

(4.26)

4.3. REAL SPACE AND EIGENSPACE

61

Note that the steady-state solution is A; ~f , as might be expected. The rst group of terms on the right side of this equation is referred to classically as the complementary solution or the solution of the homogeneous equations. The second group is referred to classically as the particular solution or the particular integral. In our application to uid dynamics, it is more instructive to refer to these groups as the transient and steady-state solutions, respectively. An alternative, but entirely equivalent, form of the solution is ~u(t) = c e1 t ~x +


+ cmem t ~xm +


+ cM eM t ~xM + A; ~f (4.27) 1

1

1

1

4.3 Real Space and Eigenspace 4.3.1 De nition

Following the semi-discrete approach discussed in Section 4.1, we reduce the partial dierential equations to a set of ordinary dierential equations represented by the generic form d~u = A~u ; ~f (4.28) dt The dependent variable ~u represents some physical quantity or quantities which relate to the problem of interest. For the model problems on which we are focusing most of our attention, the elements of A are independent of both u and t. This permits us to say a great deal about such problems and serves as the basis for this section. In particular, we can develop some very important and fundamental concepts that underly the global properties of the numerical solutions to the model problems. How these relate to the numerical solutions of more practical problems depends upon the problem and, to a much greater extent, on the cleverness of the relator. We begin by developing the concept of \spaces". That is, we identify dierent mathematical reference frames (spaces) and view our solutions from within each. In this way, we get dierent perspectives of the same solution, and this can add signi cantly to our understanding. The most natural reference frame is the physical one. We say If a solution is expressed in terms of ~u, it is said to be in real space. There is, however, another very useful frame. We saw in Sections 4.2.1 and 4.2 that pre- and post-multiplication of A by the appropriate similarity matrices transforms A into a diagonal matrix, composed, in the most general case, of Jordan blocks or, in the

62

CHAPTER 4. THE SEMI-DISCRETE APPROACH

simplest nondefective case, of scalars. Following Section 4.2 and, for simplicity, using only complete systems for our examples, we found that Eq. 4.28 had the alternative form dw~ = w~ ; ~g dt which is an uncoupled set of rst-order ODE's that can be solved independently for the dependent variable vector w~ . We say If a solution is expressed in terms of w~ , it is said to be in eigenspace (often referred to as wave space). The relations that transfer from one space to the other are:

w~ = X ; ~u ~g = X ; ~f 1 1

~u = X w~ ~f = X~g

The elements of ~u relate directly to the local physics of the problem. However, the elements of w~ are linear combinations of all of the elements of ~u, and individually they have no direct local physical interpretation. When the forcing function ~f is independent of t, the solutions in the two spaces are represented by ~u(t) = X cm em t ~xm + X ; X ; ~f and wm(t) = cm em t + 1 gm ; m = 1; 2;


; M m for real space and eigenspace, respectively. At this point we make two observations: 1

1

1. the transient portion of the solution in real space consists of a linear combination of contributions from each eigenvector, and 2. the transient portion of the solution in eigenspace provides the weighting of each eigenvector component of the transient solution in real space.

4.3.2 Eigenvalue Spectrums for Model ODE's

It is instructive to consider the eigenvalue spectrums of the ODE's formulated by central dierencing the model equations for diusion and biconvection. These model ODE's are presented in Section 4.1. Equations for the eigenvalues of the simple

4.3. REAL SPACE AND EIGENSPACE

63

tridiagonals, B (M : a; b; c) and Bp(M : a; b; c), are given in Appendix B. From Section B.1, we nd for the model diusion equation with Dirichlet boundary conditions   m   m = ;2 + 2 cos M + 1
x ! m ; 4  ; m = 1; 2;


; M (4.29) sin = 2(M + 1)
x and, from Section B.4, for the model biconvection equation  2m  ; ia m =
x sin M m = 0; 1;


; M ; 1 2

2

2

where

= ;im a

m = 0; 1;


; M ; 1

m
x m = sin
x

(4.30)

m = 0; 1;


; M ; 1

(4.31)

is the modi ed wavenumber from Section 3.5, m = m, and
x = 2=M . Notice that the diusion eigenvalues are real and negative while those representing periodic convection are all pure imaginary. The interpretation of this result plays a very important role later in our stability analysis.

4.3.3 Eigenvectors of the Model Equations

Next we consider the eigenvectors of the two model equations. These follow as special cases from the results given in Appendix B.

The Diusion Model Consider Eq. 4.4, the model ODE's for diusion. First, to help visualize the matrix structure, we present results for a simple 4-point mesh and then we give the general case. The right-hand eigenvector matrix X is given by 2 sin (x ) sin (2x ) sin (3x ) sin (4x ) 3 66 sin (x ) sin (2x ) sin (3x ) sin (4x ) 77 64 sin (x ) sin (2x ) sin (3x ) sin (4x ) 75 sin (x ) sin (2x ) sin (3x ) sin (4x ) The columns of the matrix are proportional to the eigenvectors. Recall that xj = j
x = j=(M + 1), so in general the relation ~u = X w~ can be written as

uj =

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

M X

m=1

wm sin mxj ; j = 1; 2;


; M

(4.32)

CHAPTER 4. THE SEMI-DISCRETE APPROACH

64

For the inverse, or left-hand eigenvector matrix X ; , we nd 1

2 sin (x ) 66 sin (2x ) 64 sin (3x ) sin (4x )

sin (x ) sin (2x ) sin (3x ) sin (4x )

1

2

1

2

1

2

1

2

sin (x ) sin (2x ) sin (3x ) sin (4x ) 3

3 3 3

sin (x ) 3 sin (2x ) 777 sin (3x ) 5 sin (4x ) 4

4 4 4

The rows of the matrix are proportional to the eigenvectors. In general w~ = X ; ~u gives 1

wm =

M X j =1

uj sin mxj ; m = 1; 2;


; M

(4.33)

In the eld of harmonic analysis, Eq. 4.33 represents a sine transform of the function u(x) for an M -point sample between the boundaries x = 0 and x =  with the condition u(0) = u() = 0. Similarly, Eq. 4.32 represents the sine synthesis that companions the sine transform given by Eq. 4.33. In summary, For the model diusion equation:

w~ = X ; ~u is a sine transform from real space to (sine) wave space. ~u = X w~ is a sine synthesis from wave space back to real space. 1

The Biconvection Model Next consider the model ODE's for periodic convection, Eq. 4.5. The coecient matrices for these ODE's are always circulant. For our model ODE, the right-hand eigenvectors are given by

~xm = ei j

m=M ) ;

(2

j = 0; 1;


; M ; 1 m = 0; 1;


; M ; 1

With xj = j

x = j
2=M , we can write ~u = X w~ as

uj =

MX ;1 m=0

wmeimxj ; j = 0; 1;


; M ; 1

(4.34)

4.3. REAL SPACE AND EIGENSPACE

65

For a 4-point periodic mesh, we nd the following left-hand eigenvector matrix from Appendix B.4:

2w 66 w 64 w w

1 2 3 4

3 21 1 1 1 77 1 66 1 e; i= e; i= e; i= 75 = 4 64 1 e; i= e; i= e; i= 1 e; i= e; i= e; i=

In general

2

4

4

4

6

4

4

8

4

12

4

6

4

12

18

4

4

4

32u 77 66 u 75 64 u u

1 2 3

3 77 75 = X ; ~u 1

4

MX ; 1 wm = M uj e;imxj ; m = 0; 1;


; M ; 1 j This equation is identical to a discrete Fourier transform of the periodic dependent variable ~u using an M -point sample between and including x = 0 and x = 2 ;
x. For circulant matrices, it is straightforward to establish the fact that the relation ~u = X w~ represents the Fourier synthesis of the variable w~ back to ~u. In summary, 1

=0

For any circulant system:

w~ = X ; ~u is a complex Fourier transform from real space to wave space. ~u = X w~ is a complex Fourier synthesis from wave space back to real space. 1

4.3.4 Solutions of the Model ODE's

We can now combine the results of the previous sections to write the solutions of our model ODE's.

The Diusion Equation For the diusion equation, Eq. 4.27 becomes

uj (t) = where

M X m=1

cmem t sin mxj + (A; f )j ; 1

j = 1; 2;


; M

4 sin m m = ; 2(M + 1)
x 2

2

!

(4.35)

(4.36)

CHAPTER 4. THE SEMI-DISCRETE APPROACH

66

With the modi ed wavenumber de ned as  m
x  2  m =
x sin 2 and using m = m, we can write the ODE solution as

uj (t) =

M X m=1

cme;m 2t sin m xj + (A; f )j ; 1

(4.37)

j = 1; 2;


; M

(4.38)

This can be compared with the exact solution to the PDE, Eq. 2.37, evaluated at the nodes of the grid:

uj (t) =

M X m=1

cm e;m2 t sin m xj + h(xj );

j = 1; 2;


; M

(4.39)

We see that the solutions are identical except for the steady solution and the modi ed wavenumber in the transient term. The modi ed wavenumber is an approximation to the actual wavenumber. The dierence between the modi ed wavenumber and the actual wavenumber depends on the dierencing scheme and the grid resolution. This dierence causes the various modes (or eigenvector components) to decay at rates which dier from the exact solution. With conventional dierencing schemes, low wavenumber modes are accurately represented, while high wavenumber modes (if they have signi cant amplitudes) can have large errors.

The Convection Equation For the biconvection equation, we obtain

uj (t) =

MX ;1 m=0

cmem t eimxj ;

j = 0; 1;


; M ; 1

(4.40)

where

m = ;im a

(4.41)

with the modi ed wavenumber de ned in Eq. 4.31. We can write this ODE solution as

uj (t) =

MX ;1 m=0

cme;im at eimxj ;

j = 0; 1;


; M ; 1

(4.42)

4.4. THE REPRESENTATIVE EQUATION

67

and compare it to the exact solution of the PDE, Eq. 2.26, evaluated at the nodes of the grid: MX ; uj (t) = fm (0)e;imat eim xj ; j = 0; 1;


; M ; 1 (4.43) 1

m=0

Once again the dierence appears through the modi ed wavenumber contained in m. As discussed in Section 3.5, this leads to an error in the speed with which various modes are convected, since  is real. Since the error in the phase speed depends on the wavenumber, while the actual phase speed is independent of the wavenumber, the result is erroneous numerical dispersion. In the case of non-centered dierencing, discussed in Chapter 11, the modi ed wavenumber is complex. The form of Eq. 4.42 shows that the imaginary portion of the modi ed wavenumber produces nonphysical decay or growth in the numerical solution.

4.4 The Representative Equation In Section 4.3, we pointed out that Eqs. 4.20 and 4.23 express identical results but in terms of dierent groupings of the dependent variables, which are related by algebraic manipulation. This leads to the following important concept: The numerical solution to a set of linear ODE's (in which A is not a function of t) is entirely equivalent to the solution obtained if the equations are transformed to eigenspace, solved there in their uncoupled form, and then returned as a coupled set to real space. The importance of this concept resides in its message that we can analyze timemarching methods by applying them to a single, uncoupled equation and our conclusions will apply in general. This is helpful both in analyzing the accuracy of time-marching methods and in studying their stability, topics which are covered in Chapters 6 and 7. Our next objective is to nd a \typical" single ODE to analyze. We found the uncoupled solution to a set of ODE's in Section 4.2. A typical member of the family is dwm =  w ; g (t) (4.44) m m m dt The goal in our analysis is to study typical behavior of general situations, not particular problems. For such a purpose Eq. 4.44 is not quite satisfactory. The role of m is clear; it stands for some representative eigenvalue in the original A matrix. However,

CHAPTER 4. THE SEMI-DISCRETE APPROACH

68

the question is: What should we use for gm(t) when the time dependence cannot be ignored? To answer this question, we note that, in principle, one can express any one of the forcing terms gm(t) as a nite Fourier series. For example X ;g(t) = ak eikt for which Eq. 4.44 has the exact solution:

k

X ak eikt k ik ;  From this we can extract the k'th term and replace ik with . This leads to w(t) = cet +

The Representative ODE dw = w + aet dt

(4.45)

which can be used to evaluate all manner of time-marching methods. In such evaluations the parameters  and  must be allowed to take the worst possible combination of values that might occur in the ODE eigensystem. The exact solution of the representative ODE is (for  6= ): t (4.46) w(t) = cet + ae; 

4.5 Problems 1. Consider the nite-dierence operator derived in question 1 of Chapter 3. Using this operator to approximate the spatial derivative in the linear convection equation, write the semi-discrete form obtained with periodic boundary conditions on a 5-point grid (M = 5). 2. Consider the application of the operator given in Eq. 3.52 to the 1-D diusion equation with Dirichlet boundary conditions. Write the resulting semi-discrete ODE form. Find the entries in the boundary-condition vector. 3. Write the semi-discrete form resulting from the application of second-order centered dierences to the following equation on the domain 0  x  1 with boundary conditions u(0) = 0, u(1) = 1: @u = @ u ; 6x @t @x 2

2

4.5. PROBLEMS

69

4. Consider a grid with 10 interior points spanning the domain 0  x  . For initial conditions u(x; 0) = sin(mx) and boundary conditions u(0; t) = u(; t) = 0, plot the exact solution of the diusion equation with  = 1 at t = 1 with m = 1 and m = 3. (Plot the solution at the grid nodes only.) Calculate the corresponding modi ed wavenumbers for the second-order centered operator from Eq. 4.37. Calculate and plot the corresponding ODE solutions. 5. Consider the matrix

A = ;Bp(10; ;1; 0; 1)=(2
x) corresponding to the ODE form of the biconvection equation resulting from the application of second-order central dierencing on a 10-point grid. Note that the domain is 0  x  2 and
x = 2=10. The grid nodes are given by xj = j
x; j = 0; 1; : : : 9. The eigenvalues of the above matrix A, as well as the matrices X and X ; , can be found from Appendix B.4. Using these, compute and plot the ODE solution at t = 2 for the initial condition u(x; 0) = sin x. Compare with the exact solution of the PDE. Calculate the numerical phase speed from the modi ed wavenumber corresponding to this initial condition and show that it is consistent with the ODE solution. Repeat for the initial condition u(x; 0) = sin 2x. 1

70

CHAPTER 4. THE SEMI-DISCRETE APPROACH

Chapter 5 FINITE-VOLUME METHODS In Chapter 3, we saw how to derive nite-dierence approximations to arbitrary derivatives. In Chapter 4, we saw that the application of a nite-dierence approximation to the spatial derivatives in our model PDE's produces a coupled set of ODE's. In this Chapter, we will show how similar semi-discrete forms can be derived using nite-volume approximations in space. Finite-volume methods have become popular in CFD as a result, primarily, of two advantages. First, they ensure that the discretization is conservative, i.e., mass, momentum, and energy are conserved in a discrete sense. While this property can usually be obtained using a nite-dierence formulation, it is obtained naturally from a nite-volume formulation. Second, nitevolume methods do not require a coordinate transformation in order to be applied on irregular meshes. As a result, they can be applied on unstructured meshes consisting of arbitrary polyhedra in three dimensions or arbitrary polygons in two dimensions. This increased exibility can be used to great advantage in generating grids about arbitrary geometries. Finite-volume methods are applied to the integral form of the governing equations, either in the form of Eq. 2.1 or Eq. 2.2. Consistent with our emphasis on semi-discrete methods, we will study the latter form, which is

d Z QdV + I n:FdS = Z PdV dt V (t) S (t) V (t)

(5.1)

We will begin by presenting the basic concepts which apply to nite-volume strategies. Next we will give our model equations in the form of Eq. 5.1. This will be followed by several examples which hopefully make these concepts clear. 71

72

5.1 Basic Concepts

CHAPTER 5. FINITE-VOLUME METHODS

The basic idea of a nite-volume method is to satisfy the integral form of the conservation law to some degree of approximation for each of many contiguous control volumes which cover the domain of interest. Thus the volume V in Eq. 5.1 is that of a control volume whose shape is dependent on the nature of the grid. In our examples, we will consider only control volumes which do not vary with time. Examining Eq. 5.1, we see that several approximations must be made. The ux is required at the boundary of the control volume, which is a closed surface in three dimensions and a closed contour in two dimensions. This ux must then be integrated to nd the net

ux through the boundary. Similarly, the source term P must be integrated over the control volume. Next a time-marching method1 can be applied to nd the value of Z

V

QdV

(5.2)

at the next time step. Let us consider these approximations in more detail. First, we note that the average value of Q in a cell with volume V is Z 1  Q  V QdV

V

(5.3)

and Eq. 5.1 can be written as I Z V dtd Q + S n:FdS = V PdV

(5.4)

for a control volume which does not vary with time. Thus after applying a timemarching method, we have updated values of the cell-averaged quantities Q . In order to evaluate the uxes, which are a function of Q, at the control-volume boundary, Q can be represented within the cell by some piecewise approximation which produces the correct value of Q . This is a form of interpolation often referred to as reconstruction. As we shall see in our examples, each cell will have a dierent piecewise approximation to Q. When these are used to calculate F(Q), they will generally produce dierent approximations to the ux at the boundary between two control volumes, that is, the ux will be discontinuous. A nondissipative scheme analogous to centered dierencing is obtained by taking the average of these two uxes. Another approach known as ux-dierence splitting is described in Chapter 11. The basic elements of a nite-volume method are thus the following: 1

Time-marching methods will be discussed in the next chapter.

5.2. MODEL EQUATIONS IN INTEGRAL FORM

73

1. Given the value of Q for each control volume, construct an approximation to Q(x; y; z) in each control volume. Using this approximation, nd Q at the control-volume boundary. Evaluate F(Q) at the boundary. Since there is a distinct approximation to Q(x; y; z) in each control volume, two distinct values of the ux will generally be obtained at any point on the boundary between two control volumes. 2. Apply some strategy for resolving the discontinuity in the ux at the controlvolume boundary to produce a single value of F(Q) at any point on the boundary. This issue is discussed in Section 11.4.2. 3. Integrate the ux to nd the net ux through the control-volume boundary using some sort of quadrature. 4. Advance the solution in time to obtain new values of Q . The order of accuracy of the method is dependent on each of the approximations. These ideas should be clari ed by the examples in the remainder of this chapter. In order to include diusive uxes, the following relation between rQ and Q is sometimes used: Z I rQdV = nQdS (5.5) or, in two dimensions,

V

Z

S

I

A

rQdA = C nQdl

(5.6)

where the unit vector n points outward from the surface or contour.

5.2 Model Equations in Integral Form 5.2.1 The Linear Convection Equation A two-dimensional form of the linear convection equation can be written as

@u + a cos  @u + a sin  @u = 0 @t @x @y

(5.7)

This PDE governs a simple plane wave convecting the scalar quantity, u(x; y; t) with speed a along a straight line making an angle  with respect to the x-axis. The one-dimensional form is recovered with  = 0.

CHAPTER 5. FINITE-VOLUME METHODS

74

For unit speed a, the two-dimensional linear convection equation is obtained from the general divergence form, Eq. 2.3, with

Q = u F = iu cos  + ju sin  P = 0

(5.8) (5.9) (5.10)

Since Q is a scalar, F is simply a vector. Substituting these expressions into a twodimensional form of Eq. 2.2 gives the following integral form

d Z udA + I n:(iu cos  + ju sin )ds = 0 dt A C

(5.11)

where A is the area of the cell which is bounded by the closed contour C .

5.2.2 The Diusion Equation

The integral form of the two-dimensional diusion equation with no source term and unit diusion coecient  is obtained from the general divergence form, Eq. 2.3, with

Q = u F = ;ru ! @u @u = ; i @x + j @y P = 0 Using these, we nd

(5.12) (5.13) (5.14) (5.15) !

d Z udA = I n: i @u + j @u ds dt A @x @y C

(5.16)

to be the integral form of the two-dimensional diusion equation.

5.3 One-Dimensional Examples We restrict our attention to a scalar dependent variable u and a scalar ux f , as in the model equations. We consider an equispaced grid with spacing
x. The nodes of the grid are located at xj = j
x as usual. Control volume j extends from xj ;
x=2 to xj +
x=2, as shown in Fig. 5.1. We will use the following notation:

xj;1=2 = xj ;
x=2;

xj+1=2 = xj +
x=2

(5.17)

5.3. ONE-DIMENSIONAL EXAMPLES

75 j+1/2

j-1/2 ∆ x

j-2

j-1

j L

R

j+1 L

j+2

R

Figure 5.1: Control volume in one dimension.

uj1=2 = u(xj1=2);

fj1=2 = f (uj1=2)

With these de nitions, the cell-average value becomes Z xj +1=2 1 uj (t) 
x x u(x; t)dx j ;1=2 and the integral form becomes

(5.18) (5.19)

Z xj +1=2 d (
xu ) + f ; f = Pdx (5.20) j j +1=2 j ;1=2 dt xj;1=2 Now with  = x ; xj , we can expand u(x) in Eq. 5.19 in a Taylor series about xj (with t xed) to get 2 3 ! Z
x=2 2 @2u ! 3 @3u ! @u  1 uj 
x ;
x=2 4uj +  @x + 2 @x2 + 6 @x3 + : : :5 d j j j ! ! 2 2 4 4
x @ u + O(
x6 ) = uj +
24x @@xu2 + 1920 (5.21) @x4 j j

or

uj = uj + O(
x2)

(5.22) where uj is the value at the center of the cell. Hence the cell-average value and the value at the center of the cell dier by a term of second order.

5.3.1 A Second-Order Approximation to the Convection Equation In one dimension, the integral form of the linear convection equation, Eq. 5.11, becomes (5.23)
x ddtuj + fj+1=2 ; fj;1=2 = 0

76

CHAPTER 5. FINITE-VOLUME METHODS

with f = u. We choose a piecewise constant approximation to u(x) in each cell such that u(x) = uj xj;1=2  x  xj+1=2 (5.24) Evaluating this at j + 1=2 gives fjL+1=2 = f (uLj+1=2) = uLj+1=2 = uj (5.25) where the L indicates that this approximation to fj+1=2 is obtained from the approximation to u(x) in the cell to the left of xj+1=2 , as shown in Fig. 5.1. The cell to the right of xj +1=2, which is cell j + 1, gives fjR+1=2 = uj+1 (5.26) Similarly, cell j is the cell to the right of xj;1=2 , giving fjR;1=2 = uj (5.27) and cell j ; 1 is the cell to the left of xj;1=2, giving fjL;1=2 = uj;1 (5.28) We have now accomplished the rst step from the list in Section 5.1; we have de ned the uxes at the cell boundaries in terms of the cell-average data. In this example, the discontinuity in the ux at the cell boundary is resolved by taking the average of the uxes on either side of the boundary. Thus f^j+1=2 = 21 (fjL+1=2 + fjR+1=2) = 12 (uj + uj+1) (5.29) and f^j;1=2 = 21 (fjL;1=2 + fjR;1=2) = 12 (uj;1 + uj ) (5.30) where f^ denotes a numerical ux which is an approximation to the exact ux. Substituting Eqs. 5.29 and 5.30 into the integral form, Eq. 5.23, we obtain
x ddtuj + 21 (uj + uj+1) ; 12 (uj;1 + uj ) =
x ddtuj + 12 (uj+1 ; uj;1) = 0 (5.31) With periodic boundary conditions, this point operator produces the following semidiscrete form: d~u = ; 1 B (;1; 0; 1)~u (5.32) dt 2
x p

5.3. ONE-DIMENSIONAL EXAMPLES

77

This is identical to the expression obtained using second-order centered dierences, except it is written in terms of the cell average ~u, rather than the nodal values, ~u. Hence our analysis and understanding of the eigensystem of the matrix Bp(;1; 0; 1) is relevant to nite-volume methods as well as nite-dierence methods. Since the eigenvalues of Bp(;1; 0; 1) are pure imaginary, we can conclude that the use of the average of the uxes on either side of the cell boundary, as in Eqs. 5.29 and 5.30, can lead to a nondissipative nite-volume method.

5.3.2 A Fourth-Order Approximation to the Convection Equation Let us replace the piecewise constant approximation in Section 5.3.1 with a piecewise quadratic approximation as follows u( ) = a 2 + b + c (5.33) where  is again equal to x ; xj . The three parameters a, b, and c are chosen to satisfy the following constraints: 1 Z ;
x=2 u( )d = u j ;1
x ;3
x=2 1 Z
x=2 u( )d = u j
x ;
x=2

These constraints lead to

(5.34)

1 Z 3
x=2 u( )d = u j +1
x
x=2 2uj + uj;1 a = uj+1 ;2
x2

; uj;1 b = uj+12
x

(5.35)

uj ; uj+1 c = ;uj;1 + 26 24 With these values of a, b, and c, the piecewise quadratic approximation produces

the following values at the cell boundaries: uLj+1=2 = 16 (2uj+1 + 5uj ; uj;1)

(5.36)

78

CHAPTER 5. FINITE-VOLUME METHODS uRj;1=2 = 16 (;uj+1 + 5uj + 2uj;1)

(5.37)

uRj+1=2 = 16 (;uj+2 + 5uj+1 + 2uj )

(5.38)

uLj;1=2 = 61 (2uj + 5uj;1 ; uj;2)

(5.39)

using the notation de ned in Section 5.3.1. Recalling that f = u, we again use the average of the uxes on either side of the boundary to obtain f^j+1=2 = 12 [f (uLj+1=2) + f (uRj+1=2)] 1 (;u + 7u + 7u ; u ) (5.40) = 12 j +2 j +1 j j ;1 and

f^j;1=2 = 21 [f (uLj;1=2) + f (uRj;1=2)] = 1 (;uj+1 + 7uj + 7uj;1 ; uj;2) 12

(5.41)

Substituting these expressions into the integral form, Eq. 5.23, gives 1 (;u + 8u ; 8u + u ) = 0
x ddtuj + 12 (5.42) j +2 j +1 j ;1 j ;2 This is a fourth-order approximation to the integral form of the equation, as can be veri ed using Taylor series expansions (see question 1 at the end of this chapter). With periodic boundary conditions, the following semi-discrete form is obtained: d~u = ; 1 B (1; ;8; 0; 8; ;1)~u (5.43) dt 12
x p This is a system of ODE's governing the evolution of the cell-average data.

5.3.3 A Second-Order Approximation to the Diusion Equation

In this section, we describe two approaches to deriving a nite-volume approximation to the diusion equation. The rst approach is simpler to extend to multidimensions, while the second approach is more suited to extension to higher order accuracy.

5.3. ONE-DIMENSIONAL EXAMPLES

79

In one dimension, the integral form of the diusion equation, Eq. 5.16, becomes ;f =0 (5.44)
x duj + f j +1=2

dt

j ;1=2

with f = ;ru = ;@u=@x. Also, Eq. 5.6 becomes Zb

a

@u dx = u(b) ; u(a) @x

(5.45)

We can thus write the following expression for the average value of the gradient of u over the interval xj  x  xj+1: 1 Z xj+1 @u dx = 1 (u ; u ) (5.46)
x xj @x
x j+1 j From Eq. 5.22, we know that the value of a continuous function at the center of a given interval is equal to the average value of the function over the interval to second-order accuracy. Hence, to second-order, we can write ! @u f^j+1=2 = ; @x = ; 1 (uj+1 ; uj ) (5.47)
x j +1=2 Similarly,

f^j;1=2 = ;
1x (uj ; uj;1)

(5.48)

@u = @u = 2a + b @x @

(5.51)

Substituting these into the integral form, Eq. 5.44, we obtain
x duj = 1 (uj;1 ; 2uj + uj+1) (5.49) dt
x or, with Dirichlet boundary conditions, d~u = 1 B (1; ;2; 1)~u + b~c (5.50) dt
x2 This provides a semi-discrete nite-volume approximation to the diusion equation, and we see that the properties of the matrix B (1; ;2; 1) are relevant to the study of nite-volume methods as well as nite-dierence methods. For our second approach, we use a piecewise quadratic approximation as in Section 5.3.2. From Eq. 5.33 we have

80

CHAPTER 5. FINITE-VOLUME METHODS

with a and b given in Eq. 5.35. With f = ;@u=@x, this gives fjR+1=2 = fjL+1=2 = ;
1x (uj+1 ; uj )

fjR;1=2 = fjL;1=2 = ;
1x (uj ; uj;1)

(5.52) (5.53)

Notice that there is no discontinuity in the ux at the cell boundary. This produces duj = 1 (u ; 2u + u ) (5.54) j j +1 dt
x2 j;1 which is identical to Eq. 5.49. The resulting semi-discrete form with periodic boundary conditions is d~u = 1 B (1; ;2; 1)~u (5.55) dt
x2 p which is written entirely in terms of cell-average data.

5.4 A Two-Dimensional Example The above one-dimensional examples of nite-volume approximations obscure some of the practical aspects of such methods. Thus our nal example is a nite-volume approximation to the two-dimensional linear convection equation on a grid consisting of regular triangles, as shown in Figure 5.2. As in Section 5.3.1, we use a piecewise constant approximation in each control volume and the ux at the control volume boundary is the average of the uxes obtained on either side of the boundary. The nodal data are stored at the vertices of the triangles formed by the grid. The control volumes are regular hexagons with area A,
is the length of the sides of the triangles, and ` is the length of the sides of the hexagons. The following relations hold between `,
, and A. ` = p1
p3 A = 3 2 3 `2 ` = 2 (5.56) A 3
The two-dimensional form of the conservation law is

d Z QdA + I n:Fdl = 0 dt A C

(5.57)

5.4. A TWO-DIMENSIONAL EXAMPLE

81

c

b

2 3

p

1

l

d

a

∆

4

0 5

e

f

Figure 5.2: Triangular grid. where we have ignored the source term. The contour in the line integral is composed of the sides of the hexagon. Since these sides are all straight, the unit normals can be taken outside the integral and the ux balance is given by Z 5 d Z Q dA + X n
Fdl = 0 dt A   =0

where  indexes a side of the hexagon, as shown in Figure 5.2. A list of the normals for the mesh orientation shown is given in Table 5.1. Side;  Outward Normal; n 0 1 2 3 4 5

p

(i ; 3 j)=2 pi (i + p3 j)=2 (;i + 3 j)=2 ;pi (;i ; 3 j)=2

Table 5.1. Outward normals, see Fig. 5.2. i and j are unit normals along x and y, respectively.

CHAPTER 5. FINITE-VOLUME METHODS

82

For Eq. 5.11, the two-dimensional linear convection equation, we have for side  Z

n
Fdl = n
(i cos  + j sin ) 

Z `=2

;`=2

u ( )d

(5.58)

where  is a length measured from the middle of a side  . Making the change of variable z = =`, one has the expression Z `=2

Z 1=2

u( )d = ` ;1=2 u(z)dz (5.59) Then, in terms of u and the hexagon area A, we have " Z # 5 1=2 d Z u dA + X n
(i cos  + j sin ) ` u(z)dz = 0 (5.60) dt A ;1=2  =0  The values of n
(i cos  + j sin ) are given by the expressions in Table 5.2. There ;`=2

are no numerical approximations in Eq. 5.60. That is, if the integrals in the equation are evaluated exactly, the integrated time rate of change of the integral of u over the area of the hexagon is known exactly. Side;  n
(i cos  + j sin )

p

(cos  ; 3 sin )=2 cos p (cos  + p3 sin )=2 (; cos  + 3 sin )=2 ; cos p (; cos  ; 3 sin )=2

0 1 2 3 4 5

Table 5.2. Weights of ux integrals, see Eq. 5.60. Introducing the cell average, Z

A

u dA = Aup

(5.61)

and the piecewise-constant approximation u = up over the entire hexagon, the approximation to the ux integral becomes trivial. Taking the average of the ux on either side of each edge of the hexagon gives for edge 1: Z 1=2 Z u  + u  p a dz = up + ua u (z)dz = (5.62) 2 ;1=2 2 1

5.5. PROBLEMS

83

Similarly, we have for the other ve edges: Z

u (z)dz = up +2 ub 2

Z Z

(5.63)

u (z)dz = up +2 uc 3

(5.64)

u (z)dz = up +2 ud 4

(5.65)

Z

u (z)dz = up +2 ue 5

(5.66)

u (z)dz = up +2 uf 0

(5.67)

dup + 1 [(2 cos )(u ; u ) + (cos  + p3 sin )(u ; u ) a d b e dt 3
p +(; cos  + 3 sin )(uc ; uf )] = 0

(5.69)

Z

Substituting these into Eq. 5.60, along with the expressions in Table 5.2, we obtain p A ddtup + 2` [(2 cos )(ua ; ud) + (cos  + 3 sin )(ub ; ue) p +(; cos  + 3 sin )(uc ; uf )] = 0 (5.68) or

The reader can verify, using Taylor series expansions, that this is a second-order approximation to the integral form of the two-dimensional linear convection equation.

5.5 Problems 1. Use Taylor series to verify that Eq. 5.42 is a fourth-order approximation to Eq. 5.23. 2. Find the semi-discrete ODE form governing the cell-average data resulting from the use of a linear approximation in developing a nite-volume method for the linear convection equation. Use the following linear approximation:

u( ) = a + b

CHAPTER 5. FINITE-VOLUME METHODS

84 where b = uj and

; uj;1 a = uj+12
x

and use the average ux at the cell interface. 3. Using the rst approach given in Section 5.3.3, derive a nite-volume approximation to the spatial terms in the two-dimensional diusion equation on a square grid. 4. Repeat question 3 for a grid consisting of equilateral triangles.

Chapter 6 TIME-MARCHING METHODS FOR ODE'S After discretizing the spatial derivatives in the governing PDE's (such as the NavierStokes equations), we obtain a coupled system of nonlinear ODE's in the form

d~u = F~ (~u; t) dt

(6.1)

These can be integrated in time using a time-marching method to obtain a timeaccurate solution to an unsteady ow problem. For a steady ow problem, spatial discretization leads to a coupled system of nonlinear algebraic equations in the form F~ (~u) = 0 (6.2) As a result of the nonlinearity of these equations, some sort of iterative method is required to obtain a solution. For example, one can consider the use of Newton's method, which is widely used for nonlinear algebraic equations (See Section 6.10.3.). This produces an iterative method in which a coupled system of linear algebraic equations must be solved at each iteration. These can be solved iteratively using relaxation methods, which will be discussed in Chapter 9, or directly using Gaussian elimination or some variation thereof. Alternatively, one can consider a time-dependent path to the steady state and use a time-marching method to integrate the unsteady form of the equations until the solution is suciently close to the steady solution. The subject of the present chapter, time-marching methods for ODE's, is thus relevant to both steady and unsteady ow problems. When using a time-marching method to compute steady ows, the goal is simply to remove the transient portion of the solution as quickly as possible; timeaccuracy is not required. This motivates the study of stability and stiness, topics which are covered in the next two chapters. 85

86

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

Application of a spatial discretization to a PDE produces a coupled system of ODE's. Application of a time-marching method to an ODE produces an ordinary dierence equation (O
E ). In earlier chapters, we developed exact solutions to our model PDE's and ODE's. In this chapter we will present some basic theory of linear O
E's which closely parallels that for linear ODE's, and, using this theory, we will develop exact solutions for the model O
E's arising from the application of timemarching methods to the model ODE's.

6.1 Notation Using the semi-discrete approach, we reduce our PDE to a set of coupled ODE's represented in general by Eq. 4.1. However, for the purpose of this chapter, we need only consider the scalar case

du = u0 = F (u; t) dt

(6.3)

Although we use u to represent the dependent variable, rather than w, the reader should recall the arguments made in Chapter 4 to justify the study of a scalar ODE. Our rst task is to nd numerical approximations that can be used to carry out the time integration of Eq. 6.3 to some given accuracy, where accuracy can be measured either in a local or a global sense. We then face a further task concerning the numerical stability of the resulting methods, but we postpone such considerations to the next chapter. In Chapter 2, we introduced the convention that the n subscript, or the (n) superscript, always points to a discrete time value, and h represents the time interval
t. Combining this notation with Eq. 6.3 gives u0n = Fn = F (un; tn) ; tn = nh Often we need a more sophisticated notation for intermediate time steps involving temporary calculations denoted by u~, u, etc. For these we use the notation u~0n+ = F~n+ = F (~un+; tn + h) The choice of u0 or F to express the derivative in a scheme is arbitrary. They are both commonly used in the literature on ODE's. The methods we study are to be applied to linear or nonlinear ODE's, but the methods themselves are formed by linear combinations of the dependent variable and its derivative at various time intervals. They are represented conceptually by   (6.4) un+1 = f 1 u0n+1; 0u0n; ;1u0n;1;


; 0un; ;1un;1;




6.2. CONVERTING TIME-MARCHING METHODS TO O
E'S

87

With an appropriate choice of the 0s and 0s, these methods can be constructed to give a local Taylor series accuracy of any order. The methods are said to be explicit if 1 = 0 and implicit otherwise. An explicit method is one in which the new predicted solution is only a function of known data, for example, u0n, u0n;1, un; and un;1 for a method using two previous time levels, and therefore the time advance is simple. For an implicit method, the new predicted solution is also a function of the time derivative at the new time level, that is, u0n+1. As we shall see, for systems of ODE's and nonlinear problems, implicit methods require more complicated strategies to solve for un+1 than explicit methods.

6.2 Converting Time-Marching Methods to O
E's Examples of some very common forms of methods used for time-marching general ODE's are:

un+1 = un + hu0n

(6.5)

un+1 = un + hu0n+1

(6.6)

and

u~n+1 = un + hu0n un+1 = 21 [un + u~n+1 + hu~0n+1]

(6.7)

According to the conditions presented under Eq. 6.4, the rst and third of these are examples of explicit methods. We refer to them as the explicit Euler method and the MacCormack predictor-corrector method,1 respectively. The second is implicit and referred to as the implicit (or backward) Euler method. These methods are simple recipes for the time advance of a function in terms of its value and the value of its derivative, at given time intervals. The material presented in Chapter 4 develops a basis for evaluating such methods by introducing the concept of the representative equation

du = u0 = u + aet (6.8) dt written here in terms of the dependent variable, u. The value of this equation arises from the fact that, by applying a time-marching method, we can analytically convert

1 Here we give only MacCormack's time-marching method. The method commonly referred to as

MacCormack's method, which is a fully-discrete method, will be presented in Section 11.3

88

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

such a linear ODE into a linear O
E . The latter are subject to a whole body of analysis that is similar in many respects to, and just as powerful as, the theory of ODE's. We next consider examples of this conversion process and then go into the general theory on solving O
E's. Apply the simple explicit Euler scheme, Eq. 6.5, to Eq. 6.8. There results un+1 = un + h(un + aehn) or un+1 ; (1 + h)un = haehn (6.9) Eq. 6.9 is a linear O
E , with constant coecients, expressed in terms of the dependent variable un and the independent variable n. As another example, applying the implicit Euler method, Eq. 6.6, to Eq. 6.8, we nd   un+1 = un + h un+1 + aeh(n+1) or (1 ; h)un+1 ; un = heh
aehn (6.10) As a nal example, the predictor-corrector sequence, Eq. 6.7, gives u~n+1 ; (1 + h)un = ahehn (6.11) ; 21 (1 + h)~un+1 + un+1 ; 12 un = 12 aheh(n+1) which is a coupled set of linear O
E's with constant coecients. Note that the rst line in Eq. 6.11 is identical to Eq. 6.9, since the predictor step in Eq. 6.7 is simply the explicit Euler method. The second line in Eq. 6.11 is obtained by noting that u~0n+1 = F (~un+1; tn + h) = u~n+1 + aeh(n+1) (6.12) Now we need to develop techniques for analyzing these dierence equations so that we can compare the merits of the time-marching methods that generated them.

6.3 Solution of Linear O
E's With Constant Coecients The techniques for solving linear dierence equations with constant coecients is as well developed as that for ODE's and the theory follows a remarkably parallel path. This is demonstrated by repeating some of the developments in Section 4.2, but for dierence rather than dierential equations.

6.3. SOLUTION OF LINEAR O
E'S WITH CONSTANT COEFFICIENTS

89

6.3.1 First- and Second-Order Dierence Equations First-Order Equations

The simplest nonhomogeneous O
E of interest is given by the single rst-order equation

un+1 = un + abn

(6.13)

where , a, and b are, in general, complex parameters. The independent variable is now n rather than t, and since the equations are linear and have constant coecients,  is not a function of either n or u. The exact solution of Eq. 6.13 is n un = c1n + bab; 

where c1 is a constant determined by the initial conditions. In terms of the initial value of u it can be written n ; n un = u0n + a b b ; 

Just as in the development of Eq. 4.10, one can readily show that the solution of the defective case, (b = ), un+1 = un + an is h i un = u0 + an;1 n This can all be easily veri ed by substitution.

Second-Order Equations The homogeneous form of a second-order dierence equation is given by

un+2 + a1 un+1 + a0un = 0

(6.14)

Instead of the dierential operator D  dtd used for ODE's, we use for O
E's the dierence operator E (commonly referred to as the displacement or shift operator) and de ned formally by the relations

un+1 = Eun ; un+k = E k un Further notice that the displacement operator also applies to exponents, thus

b
bn = bn+ = E 
bn

90

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

where  can be any fraction or irrational number. The roles of D and E are the same insofar as once they have been introduced to the basic equations the value of u(t) or un can be factored out. Thus Eq. 6.14 can now be re-expressed in an operational notion as (E 2 + a1E + a0)un = 0

(6.15)

which must be zero for all un. Eq. 6.15 is known as the operational form of Eq. 6.14. The operational form contains a characteristic polynomial P (E ) which plays the same role for dierence equations that P (D) played for dierential equations; that is, its roots determine the solution to the O
E. In the analysis of O
E's, we label these roots 1 , 2 ,


, etc, and refer to them as the -roots. They are found by solving the equation P () = 0. In the simple example given above, there are just two  roots and in terms of them the solution can be written

un = c1(1 )n + c2(2 )n

(6.16)

where c1 and c2 depend upon the initial conditions. The fact that Eq. 6.16 is a solution to Eq. 6.14 for all c1 ; c2 and n should be veri ed by substitution.

6.3.2 Special Cases of Coupled First-Order Equations A Complete System

Coupled, rst-order, linear homogeneous dierence equations have the form

u(1n+1) = c11u(1n) + c12 u(2n) u(2n+1) = c21u(1n) + c22 u(2n)

(6.17)

which can also be written

  ~un+1 = C~un ; ~un = hu(1n); u(2n)iT ; C = c11 c12 c21 c22

The operational form of Eq. 6.17 can be written

 (c ; E ) c12   u1 (n) = [ C ; E I ]~u = 0 11 n c21 (c22 ; E ) u2 which must be zero for all u1 and u2. Again we are led to a characteristic polynomial, this time having the form P (E ) = det [ C ; E I ]. The -roots are found from  (c ; )  c 11 12 P () = det c21 (c22 ; ) = 0

6.4. SOLUTION OF THE REPRESENTATIVE O
E'S

91

Obviously the k are the eigenvalues of C and, following the logic of Section 4.2, ~ if x are its eigenvectors, the solution of Eq. 6.17 is 2 ~un = X ck (k )n~xk k=1

where ck are constants determined by the initial conditions.

A Defective System

The solution of O
E's with defective eigensystems follows closely the logic in Section 4.2.2 for defective ODE's. For example, one can show that the solution to 2 3 2 32 3 u n+1  64 u^n+1 75 = 64 1  75 64 uu^nn 75 un+1 1  un is

un = uh0n i u^n = u^0 + u0n;1 n " # n ( n ; 1) ; 1 ; 2 un = u0 + u^0n + u0 2  n

(6.18)

6.4 Solution of the Representative O
E's 6.4.1 The Operational Form and its Solution

Examples of the nonhomogeneous, linear, rst-order ordinary dierence equations, produced by applying a time-marching method to the representative equation, are given by Eqs. 6.9 to 6.11. Using the displacement operator, E , these equations can be written

"

[E ; (1 + h)]un = h
aehn

(6.19)

[(1 ; h)E ; 1]un = h
E
aehn

(6.20)

;(1 + h) # " u~ # = h
" u n ; 21 (1 + h)E E ; 21 E

#

1 1 E
aehn 2

(6.21)

92

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

All three of these equations are subsets of the operational form of the representative O
E P (E )un = Q(E )
aehn (6.22) which is produced by applying time-marching methods to the representative ODE, Eq. 4.45. We can express in terms of Eq. 6.22 all manner of standard time-marching methods having multiple time steps and various types of intermediate predictor-corrector families. The terms P (E ) and Q(E ) are polynomials in E referred to as the characteristic polynomial and the particular polynomial, respectively. The general solution of Eq. 6.22 can be expressed as K h X un = ck (k )n + aehn
Q(eh) (6.23) P (e ) k=1 where k are the K roots of the characteristic polynomial, P () = 0. When determinants are involved in the construction of P (E ) and Q(E ), as would be the case for Eq. 6.21, the ratio Q(E )=P (E ) can be found by Kramer's rule. Keep in mind that for methods such as in Eq. 6.21 there are multiple (two in this case) solutions, one for un and u~n and we are usually only interested in the nal solution un. Notice also, the important subset of this solution which occurs when  = 0, representing a time invariant particular solution, or a steady state. In such a case K X (1) un = ck (k )n + a
PQ(1) k=1

6.4.2 Examples of Solutions to Time-Marching O
E's

As examples of the use of Eqs. 6.22 and 6.23, we derive the solutions of Eqs. 6.19 to 6.21. For the explicit Euler method, Eq. 6.19, we have P (E ) = E ; 1 ; h Q(E ) = h (6.24) and the solution of its representative O
E follows immediately from Eq. 6.23:

un = c1(1 + h)n + aehn


h

eh ; 1 ; h

For the implicit Euler method, Eq. 6.20, we have P (E ) = (1 ; h)E ; 1 Q(E ) = hE

(6.25)

6.5. THE  ;  RELATION so

93

 1 n heh un = c1 1 ; h + aehn
(1 ; h)eh ; 1

In the case of the coupled predictor-corrector equations, Eq. 6.21, one solves for the nal family un (one can also nd a solution for the intermediate family u~), and there results #  "  E ; (1 + h ) P (E ) = det ; 1 (1 + h)E E ; 1 = E E ; 1 ; h ; 12 2h2 2 2   1 E h Q(E ) = det ; 1 (1 + h)E 1 hE = 2 hE (E + 1 + h) 2 2 The -root is found from   1 2 2 P () =   ; 1 ; h ; 2  h = 0 which has only one nontrivial root ( = 0 is simply a shift in the reference index). The complete solution can therefore be written 1 heh + 1 + h   n un = c1 1 + h + 21 2h2 + aehn
2 (6.26) eh ; 1 ; h ; 12 2h2

6.5 The  ;  Relation

6.5.1 Establishing the Relation

We have now been introduced to two basic kinds of roots, the -roots and the -roots. The former are the eigenvalues of the A matrix in the ODE's found by space dierencing the original PDE, and the latter are the roots of the characteristic polynomial in a representative O
E found by applying a time-marching method to the representative ODE. There is a fundamental relation between the two which can be used to identify many of the essential properties of a time-march method. This relation is rst demonstrated by developing it for the explicit Euler method. First we make use of the semi-discrete approach to nd a system of ODE's and then express its solution in the form of Eq. 4.27. Remembering that t = nh, one can write

 n  n  n u(t) = c1 e1 h ~x1 +


+ cm em h ~xm +


+ cM eM h ~xM + P:S: (6.27)

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

94

where for the present we are not interested in the form of the particular solution (P:S:). Now the explicit Euler method produces for each -root, one -root, which is given by  = 1 + h. So if we use the Euler method for the time advance of the ODE's, the solution2 of the resulting O
E is

un = c1(1 )n ~x1 +


+ cm (m )n ~xm +


+ cM (M )n ~xM + P:S:

(6.28)

where the cm and the ~xm in the two equations are identical and m = (1 + m h). Comparing Eq. 6.27 and Eq. 6.28, we see a correspondence between m and em h. Since the value of eh can be expressed in terms of the series

eh = 1 + h + 21 2h2 + 16 3h3 +


+ n1! nhn +


the truncated expansion  = 1 + h is a reasonable3 approximation for small enough h. Suppose, instead of the Euler method, we use the leapfrog method for the time advance, which is de ned by

un+1 = un;1 + 2hu0n

(6.29)

Applying Eq. 6.8 to Eq. 6.29, we have the characteristic polynomial P (E ) = E 2 ; 2hE ; 1, so that for every  the  must satisfy the relation

m2 ; 2m hm ; 1 = 0

(6.30)

Now we notice that each  produces two -roots. For one of these we nd

q m = mh + 1 + 2mh2 = 1 + mh + 21 2m h2 ; 18 4mh4 +




(6.31) (6.32)

 h 3 3 This q is 2an2 approximation to e m with an error O( h ). The other root, mh ; 1 + mh , will be discussed in Section 6.5.3. 2 Based on Section 4.4. 3 The error is O(2 h2 ).

6.5. THE  ;  RELATION

95

6.5.2 The Principal -Root

Based on the above we make the following observation: Application of the same time-marching method to all of the equations in a coupled system linear ODE's in the form of Eq. 4.6, always produces one -root for every -root that satis es the relation (6.33)  k+1 1 1 2 2 k k  = 1 + h + 2  h +


+ k!  h + O h where k is the order of the time-marching method.

We refer to the root that has the above property as the principal -root, and designate it (m )1 . The above property can be stated regardless of  kthe  details of the time+1 marching method, knowing only that its leading error is O h . Thus the principal  root is an approximation to eh up to O hk . Note that a second-order approximation to a derivative written in the form (t u)n = 1 (un+1 ; un;1) (6.34) 2h has a leading truncation error which is O(h2), while the second-order time-marching method which results from this approximation, which is the leapfrog method:

un+1 = un;1 + 2hu0n

(6.35)

has a leading truncation error O(h3). This arises simply because of our notation for the time-marching method in which we have multiplied through by h to get an approximation for the function un+1 rather than the derivative as in Eq. 6.34. The following example makes this clear. Consider a solution obtained at a given time T using a second-order time-marching method with a time step h. Now consider the solution obtained using the same method with a time step h=2. Since the error per time step is O(h3), this is reduced by a factor of eight (considering the leading term only). However, twice as many time steps are required to reach the time T . Therefore the error at the end of the simulation is reduced by a factor of four, consistent with a second-order approximation.

6.5.3 Spurious -Roots

We saw from Eq. 6.30 that the  ;  relation for the leapfrog method produces two -roots for each . One of these we identi ed as the principal root which always

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CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

has the property given in 6.33. The other is referred to as a spurious -root and designated (m )2 . In general, the  ;  relation produced by a time-marching scheme can result in multiple -roots all of which, except for the principal one, are spurious. All spurious roots are designated (m )k where k = 2; 3;


. No matter whether a -root is principal or spurious, it is always some algebraic function of the product h. To express this fact we use the notation  = (h). If a time-marching method produces spurious -roots, the solution for the O
E in the form shown in Eq. 6.28 must be modi ed. Following again the message of Section 4.4, we have un = c11 (1 )n1 ~x1 +


+ cm1(m )n1 ~xm +


+ cM 1 (M )n1 ~xM + P:S: +c12 (1)n2 ~x1 +


+ cm2 (m )n2 ~xm +


+ cM 2 (M )n2 ~xM +c13 (1)n3 ~x1 +


+ cm3 (m )n3 ~xm +


+ cM 3 (M )n3 ~xM +etc., if there are more spurious roots (6.36) Spurious roots arise if a method uses data from time level n ; 1 or earlier to advance the solution from time level n to n + 1. Such roots originate entirely from the numerical approximation of the time-marching method and have nothing to do with the ODE being solved. However, generation of spurious roots does not, in itself, make a method inferior. In fact, many very accurate methods in practical use for integrating some forms of ODE's have spurious roots. It should be mentioned that methods with spurious roots are not self starting. For example, if there is one spurious root to a method, all of the coecients (cm )2 in Eq. 6.36 must be initialized by some starting procedure. The initial vector ~u0 does not provide enough data to initialize all of the coecients. This results because methods which produce spurious roots require data from time level n ; 1 or earlier. For example, the leapfrog method requires ~un;1 and thus cannot be started using only ~un. Presumably (i.e., if one starts the method properly) the spurious coecients are all initialized with very small magnitudes, and presumably the magnitudes of the spurious roots themselves are all less than one (see Chapter 7). Then the presence of spurious roots does not contaminate the answer. That is, after some nite time the amplitude of the error associated with the spurious roots is even smaller then when it was initialized. Thus while spurious roots must be considered in stability analysis, they play virtually no role in accuracy analysis.

6.5.4 One-Root Time-Marching Methods

There are a number of time-marching methods that produce only one -root for each -root. We refer to them as one-root methods. They are also called one-step methods.

6.6. ACCURACY MEASURES OF TIME-MARCHING METHODS

97

They have the signi cant advantage of being self-starting which carries with it the very useful property that the time-step interval can be changed at will throughout the marching process. Three one-root methods were analyzed in Section 6.4.2. A popular method having this property, the so-called -method, is given by the formula h i un+1 = un + h (1 ; )u0n + u0n+1

The -method represents the explicit Euler ( = 0), the trapezoidal ( = 21 ), and the implicit Euler methods ( = 1), respectively. Its  ;  relation is )h  = 1 +1(1; ;h It is instructive to compare the exact solution to a set of ODE's (with a complete eigensystem) having time-invariant forcing terms with the exact solution to the O
E's for one-root methods. These are ~u(t) = c1e1 hn ~x1 +


+ cm em hn ~xm +


+ cM eM hn ~xM + A;1~f ~un = c1(1 )n ~x1 +


+ cm(m )n ~xm +


+ cM (M )n ~xM + A;1~f (6.37) respectively. Notice that when t and n = 0, these equations are identical, so that all the constants, vectors, and matrices are identical except the ~u and the terms inside the parentheses on the right hand sides. The only error made by introducing the time marching is the error that  makes in approximating eh.

6.6 Accuracy Measures of Time-Marching Methods 6.6.1 Local and Global Error Measures

There are two broad categories of errors that can be used to derive and evaluate timemarching methods. One is the error made in each time step. This is a local error such as that found from a Taylor table analysis, see Section 3.4. It is usually used as the basis for establishing the order of a method. The other is the error determined at the end of a given event which has covered a speci c interval of time composed of many time steps. This is a global error. It is useful for comparing methods, as we shall see in Chapter 8. It is quite common to judge a time-marching method on the basis of results found from a Taylor table. However, a Taylor series analysis is a very limited tool for nding the more subtle properties of a numerical time-marching method. For example, it is of no use in:

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

98

 nding spurious roots.  evaluating numerical stability and separating the errors in phase and amplitude.  analyzing the particular solution of predictor-corrector combinations.  nding the global error. The latter three of these are of concern to us here, and to study them we make use of the material developed in the previous sections of this chapter. Our error measures are based on the dierence between the exact solution to the representative ODE, given by t u(t) = cet + ae; 

(6.38)

and the solution to the representative O
E's, including only the contribution from the principal root, which can be written as h

un = c1 (1)n + aehn
Q(eh ) P (e )

(6.39)

6.6.2 Local Accuracy of the Transient Solution (er;  j

Transient error

j

; er! )

The particular choice of an error measure, either local or global, is to some extent arbitrary. However, a necessary condition for the choice should be that the measure can be used consistently for all methods. In the discussion of the - relation we saw that all time-marching methods produce a principal -root for every -root that exists in a set of linear ODE's. Therefore, a very natural local error measure for the transient solution is the value of the dierence between solutions based on these two roots. We designate this by er and make the following de nition

er  eh ; 1 The leading error term can be found by expanding in a Taylor series and choosing the rst nonvanishing term. This is similar to the error found from a Taylor table. The order of the method is the last power of h matched exactly.

6.6. ACCURACY MEASURES OF TIME-MARCHING METHODS

99

Amplitude and Phase Error

Suppose a  eigenvalue is imaginary. Such can indeed be the case when we study the equations governing periodic convection which produces harmonic motion. For such cases it is more meaningful to express the error in terms of amplitude and phase. Let  = i! where ! is a real number representing a frequency. Then the numerical method must produce a principal -root that is complex and expressible in the form 1 = r + ii  ei!h (6.40) From this it follows that the local error in amplitude is measured by the deviation of j1 j from unity, that is

q era = 1 ; j1 j = 1 ; (1)2r + (1 )2i

and the local error in phase can be de ned as

er!  !h ; tan;1 [(1 )i=(1 )r )]

(6.41) Amplitude and phase errors are important measures of the suitability of time-marching methods for convection and wave propagation phenomena. The approach to error analysis described in Section 3.5 can be extended to the combination of a spatial discretization and a time-marching method applied to the linear convection equation. The principal root, 1(h), is found using  = ;ia , where  is the modi ed wavenumber of the spatial discretization. Introducing the Courant number, Cn = ah=
x, we have h = ;iCn
x. Thus one can obtain values of the principal root over the range 0  
x   for a given value of the Courant number. The above expression for er! can be normalized to give the error in the phase speed, as follows ;1 ! = 1 + tan [(1)i =(1)r )] (6.42) erp = er !h Cn
x where ! = ;a. A positive value of erp corresponds to phase lag (the numerical phase speed is too small), while a negative value corresponds to phase lead (the numerical phase speed is too large).

6.6.3 Local Accuracy of the Particular Solution (er)

The numerical error in the particular solution is found by comparing the particular solution of the ODE with that for the O
E. We have found these to be given by P:S:(ODE) = aet
( ;1 )

100 and

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S h

P:S:(O
E) = aet
Q(eh) P (e )

respectively. For a measure of the local error in the particular solution we introduce the de nition

(

)

P:S:(O
E) ; 1 er  h P:S: (ODE )

(6.43)

The multiplication by h converts the error from a global measure to a local one, so that the order of er and er are consistent. In order to determine the leading error term, Eq. 6.43 can be written in terms of the characteristic and particular polynomials as

n    o er =  c;o 
( ; )Q eh ; P eh

where

(6.44)

h( ; ) co = hlim !0 P eh

The value of co is a method-dependent constant that is often equal to one. If the forcing function is independent of time,  is equal to zero, and for this case, many numerical methods generate an er that is also zero. The algebra involved in nding the order of er can be quite tedious. However, this order is quite important in determining the true order of a time-marching method by the process that has been outlined. An illustration of this is given in the section on Runge-Kutta methods.

6.6.4 Time Accuracy For Nonlinear Applications

In practice, time-marching methods are usually applied to nonlinear ODE's, and it is necessary that the advertised order of accuracy be valid for the nonlinear cases as well as for the linear ones. A necessary condition for this to occur is that the local accuracies of both the transient and the particular solutions be of the same order. More precisely, a time-marching method is said to be of order k if

er = c1
(h)k1+1 er = c2
(h)k2+1 where k = smallest of(k1; k2)

(6.45) (6.46) (6.47)

6.6. ACCURACY MEASURES OF TIME-MARCHING METHODS

101

The reader should be aware that this is not sucient. For example, to derive all of the necessary conditions for the fourth-order Runge-Kutta method presented later in this chapter the derivation must be performed for a nonlinear ODE. However, the analysis based on a linear nonhomogeneous ODE produces the appropriate conditions for the majority of time-marching methods used in CFD.

6.6.5 Global Accuracy In contrast to the local error measures which have just been discussed, we can also de ne global error measures. These are useful when we come to the evaluation of time-marching methods for speci c purposes. This subject is covered in Chapter 8 after our introduction to stability in Chapter 7. Suppose we wish to compute some time-accurate phenomenon over a xed interval of time using a constant time step. We refer to such a computation as an \event". Let T be the xed time of the event and h be the chosen step size. Then the required number of time steps, is N, given by the relation

T = Nh

Global error in the transient A natural extension of er to cover the error in an entire event is given by

Er  eT ; (1 (h))N

(6.48)

Global error in amplitude and phase If the event is periodic, we are more concerned with the global error in amplitude and phase. These are given by

Era = 1 ; and

q

(1 )2r + (1)2i

"

N

(6.49)

!#

Er!  N !h ; tan;1 ((1))i 1r ; 1 = !T ; N tan [(1)i =(1)r ]

(6.50)

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CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

Global error in the particular solution Finally, the global error in the particular solution follows naturally by comparing the solutions to the ODE and the O
E. It can be measured by

  Q eh Er  ( ; )  h  ; 1 P e

6.7 Linear Multistep Methods In the previous sections, we have developed the framework of error analysis for time advance methods and have randomly introduced a few methods without addressing motivational, developmental or design issues. In the subsequent sections, we introduce classes of methods along with their associated error analysis. We shall not spend much time on development or design of these methods, since most of them have historic origins from a wide variety of disciplines and applications. The Linear Multistep Methods (LMM's) are probably the most natural extension to time marching of the space dierencing schemes introduced in Chapter 3 and can be analyzed for accuracy or designed using the Taylor table approach of Section 3.4.

6.7.1 The General Formulation When applied to the nonlinear ODE

du = u0 = F(u; t) dt

all linear multistep methods can be expressed in the general form 1 X k=1;K

k un+k = h

1 X k=1;K

k Fn+k

(6.51)

where the notation for F is de ned in Section 6.1. The methods are said to be linear because the 's and 's are independent of u and n, and they are said to be K -step because K time-levels of data are required to marching the solution one time-step, h. They are explicit if 1 = 0 and implicit otherwise. When Eq. 6.51 is applied to the representative equation, Eq. 6.8, and the result is expressed in operational form, one nds

0 1 1 0 1 1 X X @ k E k Aun = h@ k E k A(un + aehn ) k=1;K

k=1;K

(6.52)

6.7. LINEAR MULTISTEP METHODS

103

We recall from Section 6.5.2 that a time-marching method when applied to the representative equation must provide a -root, labeled 1, that approximates eh through the order of the method. The condition referred to as consistency simply means that  ! 1 as h ! 0, and it is certainly a necessary condition for the accuracy of any time marching method. We can also agree that, to be of any value in time accuracy, a method should at least be rst-order accurate, that is  ! (1 + h) as h ! 0. One can show that these conditions are met by any method represented by Eq. 6.51 if X X X k = 0 and k = (K + k ; 1)k k

k

k

Since both sides of Eq. 6.51 can be multiplied by an arbitrary constant, these methods are often \normalized" by requiring X k = 1 k

Under this condition co = 1 in Eq. 6.44.

6.7.2 Examples

There are many special explicit and implicit forms of linear multistep methods. Two well-known families of them, referred to as Adams-Bashforth (explicit) and AdamsMoulton (implicit), can be designed using the Taylor table approach of Section 3.4. The Adams-Moulton family is obtained from Eq. 6.51 with 1 = 1; 0 = ;1; k = 0; k = ;1; ;2;


(6.53) The Adams-Bashforth family has the same 's with the additional constraint that 1 = 0. The three-step Adams-Moulton method can be written in the following form un+1 = un + h( 1u0n+1 + 0 u0n + ;1 u0n;1 + ;2u0n;2) (6.54) A Taylor table for Eq. 6.54 can be generated as

un

h
u0n

h2
u00n

h3
u000n

h4
u0000 n

1 1 1 un+1 1 1 2 6 24 ;un ;1 ;h 1 u0n+1 ; 1 ; 1 ; 1 21 ; 1 16 ;h 0 u0n ; 0 0 ;1 61 ;h ;1 un;1 ; ;1 ;1 ; ;1 21 ;h ;2 u0n;2 ;(;2)0 ;2 ;(;2)1 ;2 ;(;2)2 ;2 21 ;(;2)3 ;2 16 Table 6.1. Taylor table for the Adams-Moulton three-step linear multistep method.

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

104

This leads to the linear system 2 1 1 1 1 32 3 2 1 3 66 2 0 ;2 ;4 77 66 10 77 66 1 77 64 3 0 3 12 75 64 75 = 64 1 75 (6.55) ;1 4 0 ;4 ;32 ;2 1 to solve for the 's, resulting in 1 = 9=24; 0 = 19=24; ;1 = ;5=24; ;2 = 1=24 (6.56) which produces a method which is fourth-order accurate.4 With 1 = 0 one obtains 2 32 3 2 3 1 1 1 0 64 0 ;2 ;4 75 64 ;1 75 = 64 11 75 (6.57) 0 3 12 ;2 1 giving 0 = 23=12; ;1 = ;16=12; ;2 = 5=12 (6.58) This is the third-order Adams-Bashforth method. A list of simple methods, some of which are very common in CFD applications, is given below together with identifying names that are sometimes associated with them. In the following material AB(n) and AM(n) are used as abbreviations for the (n)th order Adams-Bashforth and (n)th order Adams-Moulton methods. One can verify that the Adams type schemes given below satisfy Eqs. 6.55 and 6.57 up to the order of the method.

Explicit Methods

un+1 un+1 un+1 un+1

= un + hu0n Euler 0 = un;1 + 2hhun Leapfrog i 1 0 0 = un + 2 h 3un ; un;1 AB2 h i h 23u0n ; 16u0n;1 + 5u0n;2 = un + 12 AB3

Implicit Methods un+1 un+1 un+1 un+1

= un + hu0nh+1 Implicit Euler i 1 = unh + 2 h u0n + u0n+1 i Trapezoidal (AM2) 1 0 = 3 4un ; un;1 + 2hun+1 h 0 i 2nd-order Backward h 0 0 = un + 12 5un+1 + 8un ; un;1 AM3

4 Recall from Section 6.5.2 that a kth-order time-marching method has a leading truncation error term which is O(hk+1 ).

6.7. LINEAR MULTISTEP METHODS

105

6.7.3 Two-Step Linear Multistep Methods

High resolution CFD problems usually require very large data sets to store the spatial information from which the time derivative is calculated. This limits the interest in multistep methods to about two time levels. The most general two-step linear multistep method (i.e., K=2 in Eq. 6.51), that is at least rst-order accurate, can be written as h i (1 +  )un+1 = [(1 + 2 )un ; un;1] + h u0n+1 + (1 ;  + ')u0n ; 'u0n;1 (6.59) Clearly the methods are explicit if  = 0 and implicit otherwise. A list of methods contained in Eq. 6.59 is given in Table 6.2. Notice that the Adams methods have  = 0, which corresponds to ;1 = 0 in Eq. 6.51. Methods with  = ;1=2, which corresponds to 0 = 0 in Eq. 6.51, are known as Milne methods.





'

0 1

0 0 0 1=2 0 ;1=2 ;1=2 ;1=6 ;1=2 0 ;5=6 ;1=6 0 ;1=2

0 0 0 0 ;1=4 ;1=3 ;1=2 ;2=9 0 1=2 ;1=3 0 1=12 ;1=6

1=2 1 3=4 1=3 1=2 5=9 0 0 0 1=3 5=12 1=6

Method Euler Implicit Euler Trapezoidal or AM2 2nd Order Backward Adams type Lees Type Two{step trapezoidal A{contractive Leapfrog AB2 Most accurate explicit Third{order implicit AM3 Milne

Order 1 1 2 2 2 2 2 2 2 2 3 3 3 4

Table 6.2. Some linear one- and two-step methods, see Eq. 6.59. One can show after a little algebra that both er and er are reduced to 0(h3) (i.e., the methods are 2nd-order accurate) if ' =  ;  + 21 The class of all 3rd-order methods is determined by imposing the additional constraint  = 2 ; 56 Finally a unique fourth-order method is found by setting  = ;' = ;=3 = 61 .

106

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

6.8 Predictor-Corrector Methods

There are a wide variety of predictor-corrector schemes created and used for a variety of purposes. Their use in solving ODE's is relatively easy to illustrate and understand. Their use in solving PDE's can be much more subtle and demands concepts5 which have no counterpart in the analysis of ODE's. Predictor-corrector methods constructed to time-march linear or nonlinear ODE's are composed of sequences of linear multistep methods, each of which is referred to as a family in the solution process. There may be many families in the sequence, and usually the nal family has a higher Taylor-series order of accuracy than the intermediate ones. Their use is motivated by ease of application and increased eciency, where measures of eciency are discussed in the next two chapters. A simple one-predictor, one-corrector example is given by 0 u~n+ = un + hu h n0 i un+1 = un + h u~n+ + u0n (6.60) where the parameters ; and are arbitrary parameters to be determined. One

can analyze this sequence by applying it to the representative equation and using the operational techniques outlined in Section 6.4. It is easy to show, following the example leading to Eq. 6.26, that

h i P (E ) = E 
E ; 1 ; ( + )h ;  2h2 Q(E ) = E 
h
[ E  + +  h]

(6.61) (6.62)

Considering only local accuracy, one is led, by following the discussion in Section 6.6, to the following observations. For the method to be second-order accurate both er and er must be O(h3). For this to hold for er, it is obvious from Eq. 6.61 that

+ = 1 ;  = 21 which provides two equations for three unknowns. The situation for er requires some algebra, but it is not dicult to show using Eq. 6.44 that the same conditions also make it O(h3). One concludes, therefore, that the predictor-corrector sequence

u~n+ = un + hu0n   2 ; 1   1 1 0 un+1 = un + 2 h  u~n+ +  u0n

is a second-order accurate method for any .

5 Such as alternating direction, fractional-step, and hybrid methods.

(6.63)

6.9. RUNGE-KUTTA METHODS

107

A classical predictor-corrector sequence is formed by following an Adams-Bashforth predictor of any order with an Adams-Moulton corrector having an order one higher. The order of the combination is then equal to the order of the corrector. If the order of the corrector is (k), we refer to these as ABM(k) methods. The Adams-BashforthMoulton sequence for k = 3 is h i u~n+1 = un + 21 h 3u0n ; u0n;1 h h5~u0 + 8u0 ; u0 i un+1 = un + 12 (6.64) n+1 n n;1 Some simple, speci c, second-order accurate methods are given below. The Gazdag method, which we discuss in Chapter 8, is h i u~n+1 = un + 21 h 3~u0n ; u~0n;1 h i un+1 = un + 12 h u~0n + u~0n+1 (6.65) The Burstein method, obtained from Eq. 6.63 with  = 1=2 is u~n+1=2 = un + 21 hu0n un+1 = un + hu~0n+1=2 and, nally, MacCormack's method, presented earlier in this chapter, is u~n+1 = un + hu0n un+1 = 21 [un + u~n+1 + hu~0n+1] Note that MacCormack's method can also be written as u~n+1 = un + hu0n un+1 = un + 12 h[u0n + u~0n+1]

(6.66)

(6.67)

(6.68)

from which it is clear that it is obtained from Eq. 6.63 with  = 1.

6.9 Runge-Kutta Methods There is a special subset of predictor-corrector methods, referred to as Runge-Kutta methods,6 that produce just one -root for each -root such that (h) corresponds

6 Although implicit and multi-step Runge-Kutta methods exist, we will consider only single-step,

explicit Runge-Kutta methods here.

108

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

to the Taylor series expansion of eh out through the order of the method and then truncates. Thus for a Runge-Kutta method of order k (up to 4th order), the principal (and only) -root is given by (6.69)  = 1 + h + 21 2h2 +


+ k1! k hk It is not particularly dicult to build this property into a method, but, as we pointed out in Section 6.6.4, it is not sucient to guarantee k'th order accuracy for the solution of u0 = F (u; t) or for the representative equation. To ensure k'th order accuracy, the method must further satisfy the constraint that er = O(hk+1) (6.70) and this is much more dicult. The most widely publicized Runge-Kutta process is the one that leads to the fourth-order method. We present it below in some detail. It is usually introduced in the form k1 = hF (un; tn) k2 = hF (un + k1; tn + h) k3 = hF (un + 1k1 + 1k2; tn + 1 h) k4 = hF (un + 2k1 + 2k2 + 2 k3; tn + 2h) followed by u(tn + h) ; u(tn) = 1 k1 + 2k2 + 3k3 + 4k4 (6.71) However, we prefer to present it using predictor-corrector notation. Thus, a scheme entirely equivalent to 6.71 is ubn+ = un + hu0n u~n+1 = un + 1 hu0n + 1hub0n+ un+2 = un + 2 hu0n + 2hub0n+ + 2hu~0n+1 (6.72) un+1 = un + 1hu0n + 2hub0n+ + 3hu~0n+1 + 4hu0n+2 Appearing in Eqs. 6.71 and 6.72 are a total of 13 parameters which are to be determined such that the method is fourth-order according to the requirements in Eqs. 6.69 and 6.70. First of all, the choices for the time samplings, , 1, and 2 , are not arbitrary. They must satisfy the relations  = 1 = 1 + 1 2 = 2 + 2 + 2 (6.73)

6.9. RUNGE-KUTTA METHODS

109

The algebra involved in nding algebraic equations for the remaining 10 parameters is not trivial, but the equations follow directly from nding P (E ) and Q(E ) and then satisfying the conditions in Eqs. 6.69 and 6.70. Using Eq. 6.73 to eliminate the 's we nd from Eq. 6.69 the four conditions

1 + 2 + 3 + 4 2 + 31 + 42 3 1 + 4( 2 + 12 ) 4 12

= 1 = 1=2 = 1=6 = 1=24

(1) (2) (3) (4)

(6.74)

These four relations guarantee that the ve terms in  exactly match the rst 5 terms in the expansion of eh . To satisfy the condition that er = O(k5), we have to ful ll four more conditions 22 + 3 12 + 422 = 1=3 (3) 23 + 3 13 + 423 = 1=4 (4) (6.75) 2 2 2 3 1 + 4( 2 + 1 2) = 1=12 (4) 31 1 + 42( 2 + 1 2 ) = 1=8 (4) The number in parentheses at the end of each equation indicates the order that is the basis for the equation. Thus if the rst 3 equations in 6.74 and the rst equation in 6.75 are all satis ed, the resulting method would be third-order accurate. As discussed in Section 6.6.4, the fourth condition in Eq. 6.75 cannot be derived using the methodology presented here, which is based on a linear nonhomogenous representative ODE. A more general derivation based on a nonlinear ODE can be found in several books.7 There are eight equations in 6.74 and 6.75 which must be satis ed by the 10 unknowns. Since the equations are overdetermined, two parameters can be set arbitrarily. Several choices for the parameters have been proposed, but the most popular one is due to Runge. It results in the \standard" fourth-order Runge-Kutta method expressed in predictor-corrector form as ubn+1=2 = un + 21 hu0n u~n+1=2 = un + 12 hub0n+1=2 un+1 = un + hu~0n+1=2 h   i un+1 = un + 16 h u0n + 2 ub0n+1=2 + u~0n+1=2 + u0n+1 (6.76)

7 The present approach based on a linear inhomogeneous equation provides all of the necessary

conditions for Runge-Kutta methods of up to third order.

110

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

Notice that this represents the simple sequence of conventional linear multistep methods referred to, respectively, as Euler Predictor Euler Corrector Leapfrog Predictor Milne Corrector

9 > > =  RK 4 > > ;

One can easily show that both the Burstein and the MacCormack methods given by Eqs. 6.66 and 6.67 are second-order Runge-Kutta methods, and third-order methods can be derived from Eqs. 6.72 by setting 4 = 0 and satisfying only Eqs. 6.74 and the rst equation in 6.75. It is clear that for orders one through four, RK methods of order k require k evaluations of the derivative function to advance the solution one time step. We shall discuss the consequences of this in Chapter 8. Higher-order RungeKutta methods can be developed, but they require more derivative evaluations than their order. For example, a fth-order method requires six evaluations to advance the solution one step. In any event, storage requirements reduce the usefulness of Runge-Kutta methods of order higher than four for CFD applications.

6.10 Implementation of Implicit Methods We have presented a wide variety of time-marching methods and shown how to derive their  ;  relations. In the next chapter, we will see that these methods can have widely dierent properties with respect to stability. This leads to various tradeos which must be considered in selecting a method for a speci c application. Our presentation of the time-marching methods in the context of a linear scalar equation obscures some of the issues involved in implementing an implicit method for systems of equations and nonlinear equations. These are covered in this Section.

6.10.1 Application to Systems of Equations Consider rst the numerical solution of our representative ODE

u0 = u + aet

(6.77)

using the implicit Euler method. Following the steps outlined in Section 6.2, we obtained (1 ; h)un+1 ; un = heh
aehn

(6.78)

6.10. IMPLEMENTATION OF IMPLICIT METHODS

111

Solving for un+1 gives

un+1 = 1 ;1h (un + heh
aehn )

(6.79)

~u0 = A~u ; f~(t)

(6.80)

This calculation does not seem particularly onerous in comparison with the application of an explicit method to this ODE, requiring only an additional division. Now let us apply the implicit Euler method to our generic system of equations given by where ~u and f~ are vectors and we still assume that A is not a function of ~u or t. Now the equivalent to Eq. 6.78 is (I ; hA)~un+1 ; ~un = ;hf~(t + h)

(6.81)

~un+1 = (I ; hA);1 [~un ; hf~(t + h)]

(6.82)

or The inverse is not actually performed, but rather we solve Eq. 6.81 as a linear system of equations. For our one-dimensional examples, the system of equations which must be solved is tridiagonal (e.g., for biconvection, A = ;aBp (;1; 0; 1)=2
x), and hence its solution is inexpensive, but in multidimensions the bandwidth can be very large. In general, the cost per time step of an implicit method is larger than that of an explicit method. The primary area of application of implicit methods is in the solution of sti ODE's, as we shall see in Chapter 8.

6.10.2 Application to Nonlinear Equations Now consider the general nonlinear scalar ODE given by

du = F (u; t) dt

(6.83)

un+1 = un + hF (un+1; tn+1)

(6.84)

Application of the implicit Euler method gives

This is a nonlinear dierence equation. As an example, consider the nonlinear ODE du + 1 u2 = 0 (6.85) dt 2

112

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

solved using implicit Euler time marching, which gives un+1 + h 21 u2n+1 = un (6.86) which requires a nontrivial method to solve for un+1. There are several dierent approaches one can take to solving this nonlinear dierence equation. An iterative method, such as Newton's method (see below), can be used. In practice, the \initial guess" for this nonlinear problem can be quite close to the solution, since the \initial guess" is simply the solution at the previous time step, which implies that a linearization approach may be quite successful. Such an approach is described in the next Section.

6.10.3 Local Linearization for Scalar Equations General Development

Let us start the process of local linearization by considering Eq. 6.83. In order to implement the linearization, we expand F (u; t) about some reference point in time. Designate the reference value by tn and the corresponding value of the dependent variable by un. A Taylor series expansion about these reference quantities gives

! ! @F @F F (u; t) = F (un; tn) + @u (u ; un) + @t (t ; tn) n n 2F ! 2F ! 1 @ @ + 2 @u2 (u ; un)2 + @u@t (u ; un)(t ; tn) n !n 2 + 21 @@tF2 (t ; tn)2 +


n

(6.87)

On the other hand, the expansion of u(t) in terms of the independent variable t is

! 2u ! @u 1 @ 2 u(t) = un + (t ; tn) @t + 2 (t ; tn) @t2 +


n n

(6.88)

If t is within h of tn , both (t ; tn)k and (u ; un)k are O(hk ), and Eq. 6.87 can be written

! ! @F @F F (u; t) = Fn + @u (u ; un) + @t (t ; tn) + O(h2) n n

(6.89)

6.10. IMPLEMENTATION OF IMPLICIT METHODS

113

Notice that this is an expansion of the derivative of the function. Thus, relative to the order of expansion of the function, it represents a second-order-accurate, locally-linear approximation to F (u; t) that is valid in the vicinity of the reference station tn and the corresponding un = u(tn). With this we obtain the locally (in the neighborhood of tn) time-linear representation of Eq. 6.83, namely

! ! ! ! du = @F u + F ; @F u + @F (t ; t ) + O(h2) n n dt @u n @u n n @t n

(6.90)

Implementation of the Trapezoidal Method As an example of how such an expansion can be used, consider the mechanics of applying the trapezoidal method for the time integration of Eq. 6.83. The trapezoidal method is given by

un+1 = un + 21 h[Fn+1 + Fn] + hO(h2)

(6.91)

where we write hO(h2) to emphasize that the method is second order accurate. Using Eq. 6.89 to evaluate Fn+1 = F (un+1; tn+1), one nds

"

!

!

#

@F + O(h2) + F un+1 = un + 12 h Fn + @F ( u n+1 ; un ) + h n @u n @t n +hO(h2) (6.92) Note that the O(h2) term within the brackets (which is due to the local linearization) is multiplied by h and therefore is the same order as the hO(h2) error from the Trapezoidal Method. The use of local time linearization updated at the end of each time step, and the trapezoidal time march, combine to make a second-order-accurate numerical integration process. There are, of course, other second-order implicit timemarching methods that can be used. The important point to be made here is that local linearization updated at each time step has not reduced the order of accuracy of a second-order time-marching process. A very useful reordering of the terms in Eq. 6.92 results in the expression "

!#

!

1 2 @F
u (6.93) 1 ; 21 h @F n = hFn + h @u n 2 @t n which is now in the delta form which will be formally introduced in Section 12.6. In many uid mechanic applications the nonlinear function F is not an explicit function

114

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

of t. In such cases the partial derivative of F (u) with respect to t is zero and Eq. 6.93 simpli es to the second-order accurate expression

"

1 ; 21 h @F @u

!# n


un = hFn

(6.94)

Notice that the RHS is extremely simple. It is the product of h and the RHS of the basic equation evaluated at the previous time step. In this example, the basic equation was the simple scalar equation 6.83, but for our applications, it is generally the space-dierenced form of the steady-state equation of some uid ow problem. A numerical time-marching procedure using Eq. 6.94 is usually implemented as follows: 1. Solve for the elements of hF~n, store them in an array say R~ , and save ~un. 2. Solve for the elements of the matrix multiplying
~un and store in some appropriate manner making use of sparseness or bandedness of the matrix if possible. Let this storage area be referred to as B . 3. Solve the coupled set of linear equations

B
~un = R~ for
~un. (Very seldom does one nd B ;1 in carrying out this step). 4. Find ~un+1 by adding
~un to ~un, thus ~un+1 =
~un + ~un The solution for ~un+1 is generally stored such that it overwrites the value of ~un and the process is repeated.

Implementation of the Implicit Euler Method We have seen that the rst-order implicit Euler method can be written

un+1 = un + hFn+1

(6.95)

if we introduce Eq. 6.90 into this method, rearrange terms, and remove the explicit dependence on time, we arrive at the form

"

1 ; h @F @u

!# n


un = hFn

(6.96)

6.10. IMPLEMENTATION OF IMPLICIT METHODS

115

We see that the only dierence between the implementation of the trapezoidal method and the implicit Euler method is the factor of 21 in the brackets of the left side of Eqs. 6.94 and 6.96. Omission of this factor degrades the method in time accuracy by one order of h. We shall see later that this method is an excellent choice for steady problems.

Newton's Method

Consider the limit h ! 1 of Eq. 6.96 obtained by dividing both sides by h and setting 1=h = 0. There results or

! @F ; @u
un = Fn n

un+1 = un ;

"

@F @u

! #;1 n

(6.97)

Fn

(6.98)

This is the well-known Newton method for nding the roots of a nonlinear equation F (u) = 0. The fact that it has quadratic convergence is veri ed by a glance at Eqs. 6.87 and 6.88 (remember the dependence on t has been eliminated for this case). By quadratic convergence, we mean that the error after a given iteration is proportional to the square of the error at the previous iteration, where the error is the dierence between the current solution and the converged solution. Quadratic convergence is thus a very powerful property. Use of a nite value of h in Eq. 6.96 leads to linear convergence, i.e., the error at a given iteration is some multiple of the error at the previous iteration. The reader should ponder the meaning of letting h ! 1 for the trapezoidal method, given by Eq. 6.94.

6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations In order to present this concept, let us consider an example involving some simple boundary-layer equations. We choose the Falkner-Skan equations from classical boundary-layer theory. Our task is to apply the implicit trapezoidal method to the equations

0

!21 df A = 0 (6.99) dt Here f represents a dimensionless stream function, and is a scaling factor. d3f + f d2f + @1 ; dt3 dt2

116

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

First of all we reduce Eq. 6.99 to a set of rst-order nonlinear equations by the transformations 2

u1 = ddtf2 ; u2 = df dt ; u3 = f

This gives the coupled set of three nonlinear equations   u01 = F1 = ;u1 u3 ; 1 ; u22 u02 = F2 = u1 u03 = F3 = u2 and these can be represented in vector notation as

(6.100)

(6.101)

d~u = F~ (~u) dt

(6.102)

A = (aij ) = @Fi =@uj

(6.103)

Now we seek to make the same local expansion that derived Eq. 6.90, except that this time we are faced with a nonlinear vector function, rather than a simple nonlinear scalar function. The required extension requires the evaluation of a matrix, called the Jacobian matrix.8 Let us refer to this matrix as A. It is derived from Eq. 6.102 by the following process For the general case involving a third order matrix this is

2 66 66 A = 666 66 4

@F1 @u1 @F2 @u1 @F3 @u1

@F1 @u2 @F2 @u2 @F3 @u2

@F1 @u3 @F2 @u3 @F3 @u3

3 77 77 77 77 75

(6.104)

The expansion of F~ (~u) about some reference state ~un can be expressed in a way similar to the scalar expansion given by eq 6.87. Omitting the explicit dependency on the independent variable t, and de ning F~ n as F~ (~un), one has 9 2.2

8 Recall that we derived the Jacobian matrices for the two-dimensional Euler equations in Section 9 The Taylor series expansion of a vector contains a vector for the rst term, a matrix times a

vector for the second term, and tensor products for the terms of higher order.

6.11. PROBLEMS

117

  F~ (~u) = F~ n + An ~u ; ~un + O(h2) (6.105) where t ; tn and the argument for O(h2) is the same as in the derivation of Eq. 6.88. Using this we can write the local linearization of Eq. 6.102 as

d~u = A ~u + F~ ; A ~u  +O(h2) n n n n dt | {z }

(6.106)

\constant" which is a locally-linear, second-order-accurate approximation to a set of coupled nonlinear ordinary dierential equations that is valid for t  tn + h. Any rst- or second-order time-marching method, explicit or implicit, could be used to integrate the equations without loss in accuracy with respect to order. The number of times, and the manner in which, the terms in the Jacobian matrix are updated as the solution proceeds depends, of course, on the nature of the problem. Returning to our simple boundary-layer example, which is given by Eq. 6.101, we nd the Jacobian matrix to be 2 3 ; u 2 u2 ;u1 3 0 0 75 A = 64 1 (6.107) 0 1 0 The student should be able to derive results for this example that are equivalent to those given for the scalar case in Eq. 6.93. Thus for the Falkner-Skan equations the trapezoidal method results in 2 h (u3)n ; h(u2)n h (u1)n 32 (
u ) 3 2 ; (u u ) ; (1 ; u2) 3 1 + 1 3n 2 2 66 776 1 n 7 6 2n7 h (u1)n 64 ; 2 5 1 0 754 (
u2)n 5 = h 4 (
u ) ( u ) h 3n 2n 0 ;2 1 We nd ~un+1 from
~un + ~un, and the solution is now advanced one step. Re-evaluate the elements using ~un+1 and continue. Without any iterating within a step advance, the solution will be second-order-accurate in time.

6.11 Problems 1. Find an expression for the nth term in the Fibonacci series, which is given by 1; 1; 2; 3; 5; 8; : : : Note that the series can be expressed as the solution to a dierence equation of the form un+1 = un + un;1. What is u25 ? (Let the rst term given above be u0.)

118

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

2. The trapezoidal method un+1 = un + 21 h(u0n+1 + u0n) is used to solve the representative ODE. (a) What is the resulting O
E? (b) What is its exact solution? (c) How does the exact steady-state solution of the O
E compare with the exact steady-state solution of the ODE if  = 0? 3. The 2nd-order backward method is given by h i un+1 = 31 4un ; un;1 + 2hu0n+1 (a) Write the O
E for the representative equation. Identify the polynomials P (E ) and Q(E ). (b) Derive the - relation. Solve for the -roots and identify them as principal or spurious. (c) Find er and the rst two nonvanishing terms in a Taylor series expansion of the spurious root. (d) Perform a -root trace relative to the unit circle for both diusion and convection. 4. Consider the time-marching scheme given by un+1 = un;1 + 23h (u0n+1 + u0n + u0n;1) (a) Write the O
E for the representative equation. Identify the polynomials P (E ) and Q(E ). (b) Derive the  ;  relation. (c) Find er . 5. Find the dierence equation which results from applying the Gazdag predictorcorrector method (Eq. 6.65) to the representative equation. Find the - relation. 6. Consider the following time-marching method:

u~n+1=3 = un + hu0n=3 un+1=2 = un + hu~0n+1=3 =2 un+1 = un + hu0n+1=2

6.11. PROBLEMS

119

Find the dierence equation which results from applying this method to the representative equation. Find the - relation. Find the solution to the dierence equation, including the homogeneous and particular solutions. Find er and er . What order is the homogeneous solution? What order is the particular solution? Find the particular solution if the forcing term is xed. 7. Write a computer program to solve the one-dimensional linear convection equation with periodic boundary conditions and a = 1 on the domain 0  x  1. Use 2nd-order centered dierences in space and a grid of 50 points. For the initial condition, use

u(x; 0) = e;0:5[(x;0:5)=]2 with  = 0:08. Use the explicit Euler, 2nd-order Adams-Bashforth (AB2), implicit Euler, trapezoidal, and 4th-order Runge-Kutta methods. For the explicit Euler and AB2 methods, use a Courant number, ah=
x, of 0.1; for the other methods, use a Courant number of unity. Plot the solutions obtained at t = 1 compared to the exact solution (which is identical to the initial condition). 8. Repeat problem 7 using 4th-order (noncompact) dierences in space. Use only 4th-order Runge-Kutta time marching at a Courant number of unity. Show solutions at t = 1 and t = 10 compared to the exact solution. 9. Using the computer program written for problem 7, compute the solution at t = 1 using 2nd-order centered dierences in space coupled with the 4th-order Runge-Kutta method for grids of 100, 200, and 400 nodes. On a log-log scale, plot the error given by

v u M exact 2 u X u t (uj ;Muj ) j =1

where M is the number of grid nodes and uexact is the exact solution. Find the global order of accuracy from the plot. 10. Using the computer program written for problem 8, repeat problem 9 using 4th-order (noncompact) dierences in space. 11. Write a computer program to solve the one-dimensional linear convection equation with in ow-out ow boundary conditions and a = 1 on the domain 0  x  1. Let u(0; t) = sin !t with ! = 10. Run until a periodic steady state is reached which is independent of the initial condition and plot your solution

120

CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

compared with the exact solution. Use 2nd-order centered dierences in space with a 1st-order backward dierence at the out ow boundary (as in Eq. 3.69) together with 4th-order Runge-Kutta time marching. Use grids with 100, 200, and 400 nodes and plot the error vs. the number of grid nodes, as described in problem 9. Find the global order of accuracy. 12. Repeat problem 11 using 4th-order (noncompact) centered dierences. Use a third-order forward-biased operator at the in ow boundary (as in Eq. 3.67). At the last grid node, derive and use a 3rd-order backward operator (using nodes j ; 3, j ; 2, j ; 1, and j ) and at the second last node, use a 3rd-order backwardbiased operator (using nodes j ; 2, j ; 1, j , and j +1; see problem 1 in Chapter 3). 13. Using the approach described in Section 6.6.2, nd the phase speed error, erp, and the amplitude error, era , for the combination of second-order centered differences and 1st, 2nd, 3rd, and 4th-order Runge-Kutta time-marching at a Courant number of unity. Also plot the phase speed error obtained using exact integration in time, i.e., that obtained using the spatial discretization alone. Note that the required -roots for the various Runge-Kutta methods can be deduced from Eq. 6.69, without actually deriving the methods. Explain your results.


  

         

      

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σ1

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σ2

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σ3

σ3

σ2

σ1

e) Gazdag

σ1

λ h=-2

σ1

f) RK2

σ1

σ1

g) RK4

λ h=-2.8 Diffusion

ωh= 2

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Chapter 8 CHOICE OF TIME-MARCHING METHODS In this chapter we discuss considerations involved in selecting a time-marching method for a speci c application. Examples are given showing how time-marching methods can be compared in a given context. An important concept underlying much of this discussion is stiness, which is de ned in the next section.

8.1 Stiness De nition for ODE's 8.1.1 Relation to -Eigenvalues The introduction of the concept referred to as \stiness" comes about from the numerical analysis of mathematical models constructed to simulate dynamic phenomena containing widely dierent time scales. De nitions given in the literature are not unique, but fortunately we now have the background material to construct a de nition which is entirely sucient for our purposes. We start with the assumption that our CFD problem is modeled with sucient accuracy by a coupled set of ODE's producing an A matrix typi ed by Eq. 7.1. Any de nition of stiness requires a coupled system with at least two eigenvalues, and the decision to use some numerical time-marching or iterative method to solve it. The dierence between the dynamic scales in physical space is represented by the dierence in the magnitude of the eigenvalues in eigenspace. In the following discussion we concentrate on the transient part of the solution. The forcing function may also be time varying in which case it would also have a time scale. However, we assume that this scale would be adequately resolved by the chosen time-marching method, and, since this part of the ODE has no eect on the numerical stability of 149

150

CHAPTER 8. CHOICE OF TIME-MARCHING METHODS I(λ h)

Stable Region λh

R(λ h)

1

Accurate Region

Figure 8.1: Stable and accurate regions for the explicit Euler method. the homogeneous part, we exclude the forcing function from further discussion in this section. Consider now the form of the exact solution of a system of ODE's with a complete eigensystem. This is given by Eq. 6.27 and its solution using a one-root, timemarching method is represented by Eq. 6.28. For a given time step, the time integration is an approximation in eigenspace that is dierent for every eigenvector ~xm . In many numerical applications the eigenvectors associated with the small jmj are well resolved and those associated with the large jmj are resolved much less accurately, if at all. The situation is represented in the complex h plane in Fig. 8.1. In this gure the time step has been chosen so that time accuracy is given to the eigenvectors associated with the eigenvalues lying in the small circle and stability without time accuracy is given to those associated with the eigenvalues lying outside of the small circle but still inside the large circle.

The whole concept of stiness in CFD arises from the fact that we often do not need the time resolution of eigenvectors associated with the large jm j in the transient solution, although these eigenvectors must remain coupled into the system to maintain a high accuracy of the spatial resolution.

8.1. STIFFNESS DEFINITION FOR ODE'S

151

8.1.2 Driving and Parasitic Eigenvalues

For the above reason it is convenient to subdivide the transient solution into two parts. First we order the eigenvalues by their magnitudes, thus j1j  j2j 


 jM j (8.1) Then we write p M X Transient = X  t m ~ (8.2) cm e xm + cmem t ~xm Solution m =1 m = p +1 | {z } {z } | Driving Parasitic This concept is crucial to our discussion. Rephrased, it states that we can separate our eigenvalue spectrum into two groups; one [1 ! p] called the driving eigenvalues (our choice of a time-step and marching method must accurately approximate the time variation of the eigenvectors associated with these), and the other, [p+1 ! M ], called the parasitic eigenvalues (no time accuracy whatsoever is required for the eigenvectors associated with these, but their presence must not contaminate the accuracy of the complete solution). Unfortunately, we nd that, although time accuracy requirements are dictated by the driving eigenvalues, numerical stability requirements are dictated by the parasitic ones.

8.1.3 Stiness Classi cations

The following de nitions are somewhat useful. An inherently stable set of ODE's is sti if In particular we de ne the ratio

jpj  jM j

Cr = jM j = jpj and form the categories

Mildly-sti Cr < 102 Strongly-sti 103 < Cr < 105 Extremely-sti 106 < Cr < 108 Pathologically-sti 109 < Cr It should be mentioned that the gaps in the sti category de nitions are intentional because the bounds are arbitrary. It is important to notice that these de nitions make no distinction between real, complex, and imaginary eigenvalues.

152

CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

8.2 Relation of Stiness to Space Mesh Size

Many ow elds are characterized by a few regions having high spatial gradients of the dependent variables and other domains having relatively low gradient phenomena. As a result it is quite common to cluster mesh points in certain regions of space and spread them out otherwise. Examples of where this clustering might occur are at a shock wave, near an airfoil leading or trailing edge, and in a boundary layer. One quickly nds that this grid clustering can strongly aect the eigensystem of the resulting A matrix. In order to demonstrate this, let us examine the eigensystems of the model problems given in Section 4.3.2. The simplest example to discuss relates to the model diusion equation. In this case the eigenvalues are all real, negative numbers that automatically obey the ordering given in Eq. 8.1. Consider the case when all of the eigenvalues are parasitic, i.e., we are interested only in the converged steady-state solution. Under these conditions, the stiness is determined by the ratio M =1. A simple calculation shows that  2  4  4  
x 2  ; = ; 1 = ;
x2 sin 2(M + 1)
x2 2   4  2 M  ;
x2 sin 2 = ;
4x2 and the ratio is 2  4 M + 1 M =1 
x2 = 4  The most important information found from this example is the fact that the stiness of the transient solution is directly related to the grid spacing. Furthermore, in diusion problems this stiness is proportional to the reciprocal of the space mesh size squared. For a mesh size M = 40, this ratio is about 680. Even for a mesh of this moderate size the problem is already approaching the category of strongly sti. For the biconvection model a similar analysis shows that jM j = j1j 
1x Here the stiness parameter is still space-mesh dependent, but much less so than for diusion-dominated problems. We see that in both cases we are faced with the rather annoying fact that the more we try to increase the resolution of our spatial gradients, the stier our equations tend to become. Typical CFD problems without chemistry vary between the mildly and strongly sti categories, and are greatly aected by the resolution of a boundary layer since it is a diusion process. Our brief analysis has been limited to equispaced !

8.3. PRACTICAL CONSIDERATIONS FOR COMPARING METHODS

153

problems, but in general the stiness of CFD problems is proportional to the mesh intervals in the manner shown above where the critical interval is the smallest one in the physical domain.

8.3 Practical Considerations for Comparing Methods We have presented relatively simple and reliable measures of stability and both the local and global accuracy of time-marching methods. Since there are an endless number of these methods to choose from, one can wonder how this information is to be used to pick a \best" choice for a particular problem. There is no unique answer to such a question. For example, it is, among other things, highly dependent upon the speed, capacity, and architecture of the available computer, and technology in uencing this is undergoing rapid and dramatic changes as this is being written. Nevertheless, if certain ground rules are agreed upon, relevant conclusions can be reached. Let us now examine some ground rules that might be appropriate. It should then be clear how the analysis can be extended to other cases. Let us consider the problem of measuring the eciency of a time{marching method for computing, over a xed interval of time, an accurate transient solution of a coupled set of ODE's. The length of the time interval, T , and the accuracy required of the solution are dictated by the physics of the particular problem involved. For example, in calculating the amount of turbulence in a homogeneous ow, the time interval would be that required to extract a reliable statistical sample, and the accuracy would be related to how much the energy of certain harmonics would be permitted to distort from a given level. Such a computation we refer to as an event. The appropriate error measures to be used in comparing methods for calculating an event are the global ones, Era , Er and Er! , discussed in Section 6.6.5, rather than the local ones er , era , and erp discussed earlier. The actual form of the coupled ODE's that are produced by the semi-discrete approach is

d~u = F~ (~u; t) dt

At every time step we must evaluate the function F~ (~u; t) at least once. This function is usually nonlinear, and its computation usually consumes the major portion of the computer time required to make the simulation. We refer to a single calculation of the vector F~ (~u; t) as a function evaluation and denote the total number of such evaluations by Fev .

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CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

8.4 Comparing the Eciency of Explicit Methods 8.4.1 Imposed Constraints

As mentioned above, the eciency of methods can be compared only if one accepts a set of limiting constraints within which the comparisons are carried out. The follow assumptions bound the considerations made in this Section: 1. The time-march method is explicit. 2. Implications of computer storage capacity and access time are ignored. In some contexts, this can be an important consideration. 3. The calculation is to be time-accurate, must simulate an entire event which takes a total time T , and must use a constant time step size, h, so that

T = Nh where N is the total number of time steps.

8.4.2 An Example Involving Diusion Let the event be the numerical solution of

du = ;u dt

(8.3)

from t = 0 to T = ; ln(0:25) with u(0) = 1. Eq. 8.3 is obtained from our representative ODE with  = ;1, a = 0. Since the exact solution is u(t) = u(0)e;t, this makes the exact value of u at the end of the event equal to 0.25, i.e., u(T ) = 0:25. To the constraints imposed above, let us set the additional requirement

 The error in u at the end of the event, i.e., the global error, must be < 0:5%. We judge the most ecient method as the one that satis es these conditions and has the fewest number of evaluations, Fev . Three methods are compared | explicit Euler, AB2, and RK4. First of all, the allowable error constraint means that the global error in the amplitude, see Eq. 6.48, must have the property:

Er < 0:005 eT

8.4. COMPARING THE EFFICIENCY OF EXPLICIT METHODS

155

Then, since h = T=N = ; ln(0:25)=N , it follows that



1 ; (1 (ln(:25)=N ))N =:25 < 0:005

where 1 is found from the characteristic polynomials given in Table 7.1. The results shown in Table 8.1 were computed using a simple iterative procedure. Method N h 1 Fev Er Euler 193 :00718 :99282 193 :001248 worst AB2 16 :0866 :9172 16 :001137 RK4 2 :6931 :5012 8 :001195 best Table 8.1: Comparison of time-marching methods for a simple dissipation problem. In this example we see that, for a given global accuracy, the method with the highest local accuracy is the most ecient on the basis of the expense in evaluating Fev . Thus the second-order Adams-Bashforth method is much better than the rstorder Euler method, and the fourth-order Runge-Kutta method is the best of all. The main purpose of this exercise is to show the (usually) great superiority of second-order over rst-order time-marching methods.

8.4.3 An Example Involving Periodic Convection

Let us use as a basis for this example the study of homogeneous turbulence simulated by the numerical solution of the incompressible Navier-Stokes equations inside a cube with periodic boundary conditions on all sides. In this numerical experiment the function evaluations contribute overwhelmingly to the CPU time, and the number of these evaluations must be kept to an absolute minimum because of the magnitude of the problem. On the other hand, a complete event must be established in order to obtain meaningful statistical samples which are the essence of the solution. In this case, in addition to the constraints given in Section 8.4.1, we add the following:

 The number of evaluations of F~ (~u; t) is xed. Under these conditions a method is judged as best when it has the highest global accuracy for resolving eigenvectors with imaginary eigenvalues. The above constraint has led to the invention of schemes that omit the function evaluation in the corrector step of a predictor-corrector combination, leading to the so-called incomplete

CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

156

predictor-corrector methods. The presumption is, of course, that more ecient methods will result from the omission of the second function evaluation. An example is the method of Gazdag, given in Section 6.8. Basically this is composed of an AB2 predictor and a trapezoidal corrector. However, the derivative of the fundamental family is never found so there is only one evaluation required to complete each cycle. The - relation for the method is shown as entry 10 in Table 7.1. In order to discuss our comparisions we introduce the following de nitions:  Let a k-evaluation method be de ned as one that requires k evaluations of F~ (~u; t) to advance one step using that method's time interval, h.  Let K represent the total number of allowable Fev .  Let h1 be the time interval advanced in one step of a one-evaluation method. The Gazdag, leapfrog, and AB2 schemes are all 1-evaluation methods. The second and fourth order RK methods are 2- and 4-evaluation methods, respectively. For a 1evaluation method the total number of time steps, N , and the number of evaluations, K , are the same, one evaluation being used for each step, so that for these methods h = h1 . For a 2-evaluation method N = K=2 since two evaluations are used for each step. However, in this case, in order to arrive at the same time T after K evaluations, the time step must be twice that of a one{evaluation method so h = 2h1. For a 4-evaluation method the time interval must be h = 4h1 , etc. Notice that as k increases, the time span required for one application of the method increases. However, notice also that as k increases, the power to which 1 is raised to arrive at the nal destination decreases; see the Figure below. This is the key to the true comparison of time-march methods for this type of problem. 0

k=1 j   k = 2 j 2h1  k=4 j 4h1



  



 



T

uN

j [(h1 )]84 j [(2h1)]2 j [(4h1)]

Step sizes and powers of  for k-evaluation methods used to get to the same value of T if 8 evaluations are allowed. In general, after K evaluations, the global amplitude and phase error for kevaluation methods applied to systems with pure imaginary -roots can be written1 Era = 1 ; j1(ik!h1 )jK=k (8.4) 1

See Eqs. 6.38 and 6.39.

8.4. COMPARING THE EFFICIENCY OF EXPLICIT METHODS 1 )]imaginary Er! = !T ; Kk tan;1 [1[(ik!h 1 (ik!h1 )]real "

#

157 (8.5)

Consider a convection-dominated event for which the function evaluation is very time consuming. We idealize to the case where  = i! and set ! equal to one. The event must proceed to the time t = T = 10. We consider two maximum evaluation limits K = 50 and K = 100 and choose from four possible methods, leapfrog, AB2, Gazdag, and RK4. The rst three of these are one-evaluation methods and the last one is a four-evaluation method. It is not dicult to show that on the basis of local error (made in a single step) the Gazdag method is superior to the RK4 method in both amplitude and phase. For example, for !h = 0:2 the Gazdag method produces a j1 j = 0:9992276 whereas for !h = 0:8 (which must be used to keep the number of evaluations the same) the RK4 method produces a j1 j = 0:998324. However, we are making our comparisons on the basis of global error for a xed number of evaluations. First of all we see that for a one-evaluation method h1 = T=K . Using this, and the fact that ! = 1, we nd, by some rather simple calculations2 made using Eqs. 8.4 and 8.5, the results shown in Table 8.2. Notice that to nd global error the Gazdag root must be raised to the power of 50 while the RK4 root is raised only to the power of 50/4. On the basis of global error the Gazdag method is not superior to RK4 in either amplitude or phase, although, in terms of phase error (for which it was designed) it is superior to the other two methods shown.

K leapfrog AB2 Gazdag RK4 !h1 = :1 100 1:0 1:003 :995 :999 !h1 = :2 50 1:0 1:022 :962 :979 a. Amplitude, exact = 1.0. K leapfrog AB2 Gazdag RK4 !h1 = :1 100 ;:96 ;2:4 :45 :12 !h1 = :2 50 ;3:8 ;9:8 1:5 1:5 b. Phase error in degrees. Table 8.2: Comparison of global amplitude and phase errors for four methods. The 1 root for the Gazdag method can be found using a numerical root nding routine to trace the three roots in the -plane, see Fig. 7.3e. 2

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CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

Using analysis such as this (and also considering the stability boundaries) the RK4 method is recommended as a basic rst choice for any explicit time-accurate calculation of a convection-dominated problem.

8.5 Coping With Stiness 8.5.1 Explicit Methods

The ability of a numerical method to cope with stiness can be illustrated quite nicely in the complex h plane. A good example of the concept is produced by studying the Euler method applied to the representative equation. The transient solution is un = (1 + h)n and the trace of the complex value of h which makes j1 + hj = 1 gives the whole story. In this case the trace forms a circle of unit radius centered at (;1; 0) as shown in Fig. 8.1. If h is chosen so that all h in the ODE eigensystem fall inside this circle the integration will be numerically stable. Also shown by the small circle centered at the origin is the region of Taylor series accuracy. If some h fall outside the small circle but stay within the stable region, these h are sti, but stable. We have de ned these h as parasitic eigenvalues. Stability boundaries for some explicit methods are shown in Figs. 7.5 and 7.6. For a speci c example, consider the mildly sti system composed of a coupled two-equation set having the two eigenvalues 1 = ;100 and 2 = ;1. If uncoupled and evaluated in wave space, the time histories of the two solutions would appear as a rapidly decaying function in one case, and a relatively slowly decaying function in the other. Analytical evaluation of the time histories poses no problem since e;100t quickly becomes very small and can be neglected in the expressions when time becomes large. Numerical evaluation is altogether dierent. Numerical solutions, of course, depend upon [(m h)]n and no jm j can exceed one for any m in the coupled system or else the process is numerically unstable. Let us choose the simple explicit Euler method for the time march. The coupled equations in real space are represented by

u1(n) = c1 (1 ; 100h)nx11 + c2(1 ; h)nx12 + (PS )1 u2(n) = c1 (1 ; 100h)nx21 + c2(1 ; h)nx22 + (PS )2

(8.6)

We will assume that our accuracy requirements are such that sucient accuracy is obtained as long as jhj  0:1. This de nes a time step limit based on accuracy considerations of h = 0:001 for 1 and h = 0:1 for 2 . The time step limit based on stability, which is determined from 1, is h = 0:02. We will also assume that c1 = c2 = 1 and that an amplitude less than 0.001 is negligible. We rst run 66 time steps with h = 0:001 in order to resolve the 1 term. With this time step the

8.5. COPING WITH STIFFNESS

159

2 term is resolved exceedingly well. After 66 steps, the amplitude of the 1 term (i.e., (1 ; 100h)n) is less than 0.001 and that of the 2 term (i.e., (1 ; h)n) is 0.9361. Hence the 1 term can now be considered negligible. To drive the (1 ; h)n term to zero (i.e., below 0.001), we would like to change the step size to h = 0:1 and continue.

We would then have a well resolved answer to the problem throughout the entire relevant time interval. However, this is not possible because of the coupled presence of (1 ; 100h)n, which in just 10 steps at h = 0:1 ampli es those terms by  109, far outweighing the initial decrease obtained with the smaller time step. In fact, with h = 0:02, the maximum step size that can be taken in order to maintain stability, about 339 time steps have to be computed in order to drive e;t to below 0.001. Thus the total simulation requires 405 time steps.

8.5.2 Implicit Methods

Now let us re-examine the problem that produced Eq. 8.6 but this time using an unconditionally stable implicit method for the time march. We choose the trapezoidal method. Its behavior in the h plane is shown in Fig. 7.4b. Since this is also a oneroot method, we simply replace the Euler  with the trapezoidal one and analyze the result. It follows that the nal numerical solution to the ODE is now represented in real space by n 1 ; 50 h u1(n) = c1 1 + 50h x11 + c2 !n 1 ; 50 h u2(n) = c1 1 + 50h x21 + c2 !

1 ; 0:5h nx + (PS ) 1 1 + 0:5h 12 ! 1 ; 0:5h nx + (PS ) 2 1 + 0:5h 22 !

(8.7)

In order to resolve the initial transient of the term e;100t , we need to use a step size of about h = 0:001. This is the same step size used in applying the explicit Euler method because here accuracy is the only consideration and a very small step size must be chosen to get the desired resolution. (It is true that for the same accuracy we could in this case use a larger step size because this is a second-order method, but that is not the point of this exercise). After 70 time steps the 1 term has amplitude less than 0.001 and can be neglected. Now with the implicit method we can proceed to calculate the remaining part of the event using our desired step size h = 0:1 without any problem of instability, with 69 steps required to reduce the amplitude of the second term to below 0.001. In both intervals the desired solution is second-order accurate and well resolved. It is true that in the nal 69 steps one -root is [1 ; 50(0:1)]=[1 + 50(0:1)] = 0:666


, and this has no physical meaning whatsoever. However, its in uence on the coupled solution is negligible at the end of the rst 70 steps, and, since (0:666


)n < 1, its in uence in the remaining 70

160

CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

steps is even less. Actually, although this root is one of the principal roots in the system, its behavior for t > 0:07 is identical to that of a stable spurious root. The total simulation requires 139 time steps.

8.5.3 A Perspective

It is important to retain a proper perspective on a problem represented by the above example. It is clear that an unconditionally stable method can always be called upon to solve sti problems with a minimum number of time steps. In the example, the conditionally stable Euler method required 405 time steps, as compared to about 139 for the trapezoidal method, about three times as many. However, the Euler method is extremely easy to program and requires very little arithmetic per step. For preliminary investigations it is often the best method to use for mildly-sti diusion dominated problems. For re ned investigations of such problems an explicit method of second order or higher, such as Adams-Bashforth or Runge-Kutta methods, is recommended. These explicit methods can be considered as eective mildly stistable methods. However, it should be clear that as the degree of stiness of the problem increases, the advantage begins to tilt towards implicit methods, as the reduced number of time steps begins to outweigh the increased cost per time step. The reader can repeat the above example with 1 = ;10; 000, 2 = ;1, which is in the strongly-sti category. There is yet another technique for coping with certain sti systems in uid dynamic applications. This is known as the multigrid method. It has enjoyed remarkable success in many practical problems; however, we need an introduction to the theory of relaxation before it can be presented.

8.6 Steady Problems In Chapter 6 we wrote the O
E solution in terms of the principal and spurious roots as follows:

un = c11 (1 )n1 ~x1 +


+ cm1(m )n1 ~xm +


+ cM 1 (M )n1 ~xM + P:S: +c12 (1)n2 ~x1 +


+ cm2 (m )n2 ~xm +


+ cM 2 (M )n2 ~xM +c13 (1)n3 ~x1 +


+ cm3 (m )n3 ~xm +


+ cM 3 (M )n3 ~xM +etc., if there are more spurious roots

(8.8)

When solving a steady problem, we have no interest whatsoever in the transient portion of the solution. Our sole goal is to eliminate it as quickly as possible. Therefore,

8.7. PROBLEMS

161

the choice of a time-marching method for a steady problem is similar to that for a sti problem, the dierence being that the order of accuracy is irrelevant. Hence the explicit Euler method is a candidate for steady diusion dominated problems, and the fourth-order Runge-Kutta method is a candidate for steady convection dominated problems, because of their stability properties. Among implicit methods, the implicit Euler method is the obvious choice for steady problems. When we seek only the steady solution, all of the eigenvalues can be considered to be parasitic. Referring to Fig. 8.1, none of the eigenvalues are required to fall in the accurate region of the time-marching method. Therefore the time step can be chosen to eliminate the transient as quickly as possible with no regard for time accuracy. For example, when using the implicit Euler method with local time linearization, Eq. 6.96, one would like to take the limit h ! 1, which leads to Newton's method, Eq. 6.98. However, a nite time step may be required until the solution is somewhat close to the steady solution.

8.7 Problems 1. Repeat the time-march comparisons for diusion (Section 8.4.2) and periodic convection (Section 8.4.3) using 2nd- and 3rd-order Runge-Kutta methods. 2. Repeat the time-march comparisons for diusion (Section 8.4.2) and periodic convection (Section 8.4.3) using the 3rd- and 4th-order Adams-Bashforth methods. Considering the stability bounds for these methods (see problem 4 in Chapter 7) as well as their memory requirements, compare and contrast them with the 3rd- and 4th-order Runge-Kutta methods. 3. Consider the diusion equation (with  = 1) discretized using 2nd-order central dierences on a grid with 10 (interior) points. Find and plot the eigenvalues and the corresponding modi ed wavenumbers. If we use the explicit Euler timemarching method what is the maximum allowable time step if all but the rst two eigenvectors are considered parasitic? Assume that sucient accuracy is obtained as long as jhj  0:1. What is the maximum allowable time step if all but the rst eigenvector are considered parasitic?

162

CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

Chapter 9 RELAXATION METHODS In the past three chapters, we developed a methodology for designing, analyzing, and choosing time-marching methods. These methods can be used to compute the time-accurate solution to linear and nonlinear systems of ODE's in the general form d~u = F~ (~u; t) (9.1) dt which arise after spatial discretization of a PDE. Alternatively, they can be used to solve for the steady solution of Eq. 9.1, which satis es the following coupled system of nonlinear algebraic equations: F~ (~u) = 0 (9.2) In the latter case, the unsteady equations are integrated until the solution converges to a steady solution. The same approach permits a time-marching method to be used to solve a linear system of algebraic equations in the form A~x = ~b (9.3) To solve this system using a time-marching method, a time derivative is introduced as follows d~x = A~x ; ~b (9.4) dt and the system is integrated in time until the transient has decayed to a suciently low level. Following a time-dependent path to steady state is possible only if all of the eigenvalues of the matrix A (or ;A) have real parts lying in the left half-plane. Although the solution ~x = A; ~b exists as long as A is nonsingular, the ODE given by Eq. 9.4 has a stable steady solution only if A meets the above condition. 1

163

CHAPTER 9. RELAXATION METHODS

164

The common feature of all time-marching methods is that they are at least rstorder accurate. In this chapter, we consider iterative methods which are not time accurate at all. Such methods are known as relaxation methods. While they are applicable to coupled systems of nonlinear algebraic equations in the form of Eq. 9.2, our analysis will focus on their application to large sparse linear systems of equations in the form Ab~u ; ~f b = 0 (9.5) where Ab is nonsingular, and the use of the subscript b will become clear shortly. Such systems of equations arise, for example, at each time step of an implicit time-marching method or at each iteration of Newton's method. Using an iterative method, we seek to obtain rapidly a solution which is arbitrarily close to the exact solution of Eq. 9.5, which is given by ~u1 = A;b ~f b (9.6) 1

9.1 Formulation of the Model Problem 9.1.1 Preconditioning the Basic Matrix

It is standard practice in applying relaxation procedures to precondition the basic equation. This preconditioning has the eect of multiplying Eq. 9.5 from the left by some nonsingular matrix. In the simplest possible case the conditioning matrix is a diagonal matrix composed of a constant D(b). If we designate the conditioning matrix by C , the problem becomes one of solving for ~u in

CAb~u ; C~f b = 0

(9.7)

Notice that the solution of Eq. 9.7 is

~u = [CAb ]; C~f b = A;b C ; C~f b = A;b ~f b 1

1

1

1

(9.8)

which is identical to the solution of Eq. 9.5, provided C ; exists. In the following we will see that our approach to the iterative solution of Eq. 9.7 depends crucially on the eigenvalue and eigenvector structure of the matrix CAb, and, equally important, does not depend at all on the eigensystem of the basic matrix Ab . For example, there are well-known techniques for accelerating relaxation schemes if the eigenvalues of CAb are all real and of the same sign. To use these schemes, the 1

9.1. FORMULATION OF THE MODEL PROBLEM

165

conditioning matrix C must be chosen such that this requirement is satis ed. A choice of C which ensures this condition is the negative transpose of Ab. For example, consider a spatial discretization of the linear convection equation using centered dierences with a Dirichlet condition on the left side and no constraint on the right side. Using a rst-order backward dierence on the right side (as in Section 3.6), this leads to the approximation

82 > > 66 ;01 10 1 > <6 ;1 0 1 x~u = 2
1 x >66 64 > ;1 0 > : ;2

39 > 77> > 77= 77> 5> > 0 ;

3 2 77 66 ;0ua 77 ~ 66 77 u + 66 0 15 4 0 2

(9.9)

The matrix in Eq. 9.9 has eigenvalues whose imaginary parts are much larger than their real parts. It can rst be conditioned so that the modulus of each element is 1. This is accomplished using a diagonal preconditioning matrix

2 66 10 6 D = 2
x 66 0 64 0

0 1 0 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1 2

3 77 77 77 5

(9.10)

which scales each row. We then further condition with multiplication by the negative transpose. The result is

A = ;AT A = 2

2 66 ;10 10 1 66 ;1 0 66 ;1 4

1

1

32 77 66 ;10 10 1 77 66 ;1 0 77 66 154 ;1

1 0 ;1 ;1

2 66 ;10 ;20 10 1 6 = 66 1 0 ;2 0 64 1 0 ;2 1

3 77 7 1 77 1 75

1 ;2

1 0 ;1

3 77 77 7 1 75 1

(9.11)

CHAPTER 9. RELAXATION METHODS

166

If we de ne a permutation matrix P and carry out the process P T [;AT A ]P (which just reorders the elements of A and doesn't change the eigenvalues) we nd 1

1

1

1

2 66 00 66 66 0 40

1 0 0 0 1 0

0 0 0 1 0

0 1 0 0 0

32

32

2 66 ;21 ;21 1 6 1 ;2 1 = 66 64 1 ;2

3 77 77 7 1 75

0 7 6 ;1 0 1 77 66 0 77 66 0 ;2 0 1 76 1 77 66 1 0 ;2 0 1 77 66 0 75 64 1 0 ;2 1 75 64 0 1 1 ;2

0 1 0 0 0

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

3

17 0 77 0 77 0 75 0

(9.12)

1 ;1 which has all negative real eigenvalues, as given in Appendix B. Thus even when the basic matrix Ab has nearly imaginary eigenvalues, the conditioned matrix ;ATb Ab is nevertheless symmetric negative de nite (i.e., symmetric with negative real eigenvalues), and the classical relaxation methods can be applied. We do not necessarily recommend the use of ;ATb as a preconditioner; we simply wish to show that a broad range of matrices can be preconditioned into a form suitable for our analysis.

9.1.2 The Model Equations

Preconditioning processes such as those described in the last section allow us to prepare our algebraic equations in advance so that certain eigenstructures are guaranteed. In the remainder of this chapter, we will thoroughly investigate some simple equations which model these structures. We will consider the preconditioned system of equations having the form

A~ ; ~f = 0 (9.13) where A is symmetric negative de nite. The symbol for the dependent variable has been changed to  as a reminder that the physics being modeled is no longer time 2

A permutation matrix (de ned as a matrix with exactly one 1 in each row and column and has the property that P T = P ;1 ) just rearranges the rows and columns of a matrix. 2 We use a symmetric negative de nite matrix to simplify certain aspects of our analysis. Relaxation methods are applicable to more general matrices. The classical methods will usually converge if Ab is diagonally dominant, as de ned in Appendix A. 1

9.1. FORMULATION OF THE MODEL PROBLEM

167

accurate when we later deal with ODE formulations. Note that the solution of Eq. 9.13, ~ = A; ~f , is guaranteed to exist because A is nonsingular. In the notation of Eqs. 9.5 and 9.7, 1

A = CAb and ~f = C~f b (9.14) The above was written to treat the general case. It is instructive in formulating the concepts to consider the special case given by the diusion equation in one dimension with unit diusion coecient  : @u = @ u ; g(x) @t @x This has the steady-state solution 2

2

(9.15)

@ u = g(x) (9.16) @x which is the one-dimensional form of the Poisson equation. Introducing the threepoint central dierencing scheme for the second derivative with Dirichlet boundary conditions, we nd 2

2

d~u = 1 B (1; ;2; 1)~u + (b~c) ; ~g (9.17) dt
x where (b~c) contains the boundary conditions and ~g contains the values of the source term at the grid nodes. In this case 2

Ab = 1 B (1; ;2; 1)
x f~b = ~g ; (b~c) (9.18) Choosing C =
x I , we obtain B (1; ;2; 1)~ = ~f (9.19) where ~f =
x f~b. If we consider a Dirichlet boundary condition on the left side and either a Dirichlet or a Neumann condition on the right side, then A has the form   A = B 1; ~b; 1 2

2

2

CHAPTER 9. RELAXATION METHODS

168

~b = [;2; ;2;



; ;2; s]T s = ;2 or ;1 (9.20) Note that s = ;1 is easily obtained from the matrix resulting from the Neumann boundary condition given in Eq. 3.24 using a diagonal conditioning matrix. A tremendous amount of insight to the basic features of relaxation is gained by an appropriate study of the one-dimensional case, and much of the remaining material is devoted to this case. We attempt to do this in such a way, however, that it is directly applicable to two- and three-dimensional problems.

9.2 Classical Relaxation

9.2.1 The Delta Form of an Iterative Scheme

We will consider relaxation methods which can be expressed in the following delta form: h i H ~n ; ~n = A~n ; ~f (9.21) where H is some nonsingular matrix which depends upon the iterative method. The matrix H is independent of n for stationary methods and is a function of n for nonstationary ones. The iteration count is designated by the subscript n or the superscript (n). The converged solution is designated ~1 so that +1

~1 = A; ~f

(9.22)

1

9.2.2 The Converged Solution, the Residual, and the Error Solving Eq. 9.21 for ~n gives ~n = [I + H ; A]~n ; H ; ~f = G~n ; H ; ~f +1

+1

1

1

1

where

(9.23)

G  I + H; A (9.24) Hence it is clear that H should lead to a system of equations which is easy to solve, or at least easier to solve than the original system. The error at the nth iteration is de ned as ~en  ~n ; ~1 = ~n ; A; ~f (9.25) 1

1

9.2. CLASSICAL RELAXATION

169

where ~1 was de ned in Eq. 9.22. The residual at the nth iteration is de ned as ~rn  A~n ; ~f (9.26) Multiply Eq. 9.25 by A from the left, and use the de nition in Eq. 9.26. There results the relation between the error and the residual A~en ; ~rn = 0 (9.27) Finally, it is not dicult to show that ~en = G~en

(9.28)

+1

Consequently, G is referred to as the basic iteration matrix, and its eigenvalues, which we designate as m , determine the convergence rate of a method. In all of the above, we have considered only what are usually referred to as stationary processes in which H is constant throughout the iterations. Nonstationary processes in which H (and possibly C ) is varied at each iteration are discussed in Section 9.5.

9.2.3 The Classical Methods

Point Operator Schemes in One Dimension Let us consider three classical relaxation procedures for our model equation B (1; ;2; 1)~ = ~f (9.29) as given in Section 9.1.2. The Point-Jacobi method is expressed in point operator form for the one-dimensional case as h i jn = 21 jn; + jn ; fj (9.30) This operator comes about by choosing the value of jn such that together with the old values of j; and j , the j th row of Eq. 9.29 is satis ed. The Gauss-Seidel method is h i jn = 21 jn; + jn ; fj (9.31) This operator is a simple extension of the point-Jacobi method which uses the most recent update of j; . Hence the j th row of Eq. 9.29 is satis ed using the new values of j and j; and the old value of j . The method of successive overrelaxation (SOR) ( +1)

( ) 1

( ) +1

( +1)

1

+1

( +1)

( +1) 1

1

1

+1

( ) +1

CHAPTER 9. RELAXATION METHODS

170

is based on the idea that if the correction produced by the Gauss-Seidel method tends to move the solution toward ~1, then perhaps it would be better to move further in this direction. It is usually expressed in two steps as h i ~j = 21 jn; + jn ; fj h i jn = jn + ! ~j ; jn (9.32) ( +1) 1

( +1)

( ) +1

( )

( )

where ! generally lies between 1 and 2, but it can also be written in the single line jn = !2 jn; + (1 ; !)jn + !2 jn ; !2 fj (9.33) ( +1)

( +1) 1

( )

( ) +1

The General Form

The general form of the classical methods is obtained by splitting the matrix A in Eq. 9.13 into its diagonal, D, the portion of the matrix below the diagonal, L, and the portion above the diagonal, U , such that

A=L+D+U

(9.34)

Then the point-Jacobi method is obtained with H = ;D, which certainly meets the criterion that it is easy to solve. The Gauss-Seidel method is obtained with H = ;(L + D), which is also easy to solve, being lower triangular.

9.3 The ODE Approach to Classical Relaxation 9.3.1 The Ordinary Dierential Equation Formulation

The particular type of delta form given by Eq. 9.21 leads to an interpretation of relaxation methods in terms of solution techniques for coupled rst-order ODE's, about which we have already learned a great deal. One can easily see that Eq. 9.21 results from the application of the explicit Euler time-marching method (with h = 1) to the following system of ODE's: ~ H ddt = A~ ; ~f (9.35) This is equivalent to d~ = H ; C A ~ ; ~f  = H ; [A~ ; ~f ] (9.36) b b dt 1

1

9.3. THE ODE APPROACH TO CLASSICAL RELAXATION

171

In the special case where H ; A depends on neither ~u nor t, H ; ~f is also independent of t, and the eigenvectors of H ; A are linearly independent, the solution can be written as 1

1

1

~ = c e1 t~x +


+ cM eM t~xM +~1 (9.37) | {z } error where what is referred to in time-accurate analysis as the transient solution, is now referred to in relaxation analysis as the error. It is clear that, if all of the eigenvalues of H ; A have negative real parts (which implies that H ; A is nonsingular), then the system of ODE's has a steady-state solution which is approached as t ! 1, given by 1

1

1

1

~1 = A; ~f 1

(9.38)

which is the solution of Eq. 9.13. We see that the goal of a relaxation method is to remove the transient solution from the general solution in the most ecient way possible. The  eigenvalues are xed by the basic matrix in Eq. 9.36, the preconditioning matrix in 9.7, and the secondary conditioning matrix in 9.35. The  eigenvalues are xed for a given h by the choice of time-marching method. Throughout the remaining discussion we will refer to the independent variable t as \time", even though no true time accuracy is involved. In a stationary method, H and C in Eq. 9.36 are independent of t, that is, they are not changed throughout the iteration process. The generalization of this in our approach is to make h, the \time" step, a constant for the entire iteration. Suppose the explicit Euler method is used for the time integration. For this method m = 1 + mh. Hence the numerical solution after n steps of a stationary relaxation method can be expressed as (see Eq. 6.28)

~n = c ~x (1 +  h)n +


+ cm~xm (1 + mh)n +


+ cM ~xM (1 + M h)n +~1 (9.39) | {z } error The initial amplitudes of the eigenvectors are given by the magnitudes of the cm . These are xed by the initial guess. In general it is assumed that any or all of the eigenvectors could have been given an equally \bad" excitation by the initial guess, so that we must devise a way to remove them all from the general solution on an equal basis. Assuming that H ; A has been chosen (that is, an iteration process has been decided upon), the only free choice remaining to accelerate the removal of the error terms is the choice of h. As we shall see, the three classical methods have all been conditioned by the choice of H to have an optimum h equal to 1 for a stationary iteration process. 1

1

1

1

CHAPTER 9. RELAXATION METHODS

172

9.3.2 ODE Form of the Classical Methods

The three iterative procedures de ned by Eqs. 9.30, 9.31 and 9.32 obey no apparent pattern except that they are easy to implement in a computer code since all of the data required to update the value of one point are explicitly available at the time of the update. Now let us study these methods as subsets of ODE as formulated in Section 9.3.1. Insert the model equation 9.29 into the ODE form 9.35. Then ~ H ddt = B (1; ;2; 1)~ ; ~f (9.40) As a start, let us use for the numerical integration the explicit Euler method n = n + h0n (9.41) with a step size, h, equal to 1. We arrive at H (~n ; ~n) = B (1; ;2; 1)~n ; ~f (9.42) +1

+1

It is clear that the best choice of H from the point of view of matrix algebra is ;B (1; ;2; 1) since then multiplication from the left by ;B ; (1; ;2; 1) gives the correct answer in one step. However, this is not in the spirit of our study, since multiplication by the inverse amounts to solving the problem by a direct method without iteration. The constraint on H that is in keeping with the formulation of the three methods described in Section 9.2.3 is that all the elements above the diagonal (or below the diagonal if the sweeps are from right to left) are zero. If we impose this constraint and further restrict ourselves to banded tridiagonals with a single constant in each band, we are led to B (; ; !2 ; 0)(~n ; ~n) = B (1; ;2; 1)~n ; ~f (9.43) where and ! are arbitrary. With this choice of notation the three methods presented in Section 9.2.3 can be identi ed using the entries in Table 9.1. 1

+1

TABLE 9.1: VALUES OF and ! IN EQ. 9.43 THAT LEAD TO CLASSICAL RELAXATION METHODS

0 1 1

!

Method

1 Point-Jacobi 1 h i Gauss-Seidel 2= 1 + sin M + 1 Optimum SOR

Equation 6:2:3 6:2:4 6:2:5

9.4. EIGENSYSTEMS OF THE CLASSICAL METHODS

173

The fact that the values in the tables lead to the methods indicated can be veri ed by simple algebraic manipulation. However, our purpose is to examine the whole procedure as a special subset of the theory of ordinary dierential equations. In this light, the three methods are all contained in the following set of ODE's d~ = B ; (; ; 2 ; 0)B (1; ;2; 1)~ ; ~f  (9.44) dt ! and appear from it in the special case when the explicit Euler method is used for its numerical integration. The point operator that results from the use of the explicit Euler scheme is !   !h n ! !h ! ! n n n n j = 2 j; + 2 (h ; )j; ; (!h ; 1)j + 2 j ; 2 fj(9.45) 1

( +1)

( +1) 1

( ) 1

( )

( ) +1

This represents a generalization of the classical relaxation techniques.

9.4 Eigensystems of the Classical Methods The ODE approach to relaxation can be summarized as follows. The basic equation to be solved came from some time-accurate derivation Ab~u ; ~f b = 0 (9.46) This equation is preconditioned in some manner which has the eect of multiplication by a conditioning matrix C giving A~ ; ~f = 0 (9.47) An iterative scheme is developed to nd the converged, or steady-state, solution of the set of ODE's ~ H ddt = A~ ; ~f (9.48) This solution has the analytical form ~n = ~en + ~1 (9.49) where ~en is the transient, or error, and ~1  A; ~f is the steady-state solution. The three classical methods, Point-Jacobi, Gauss-Seidel, and SOR, are identi ed for the one-dimensional case by Eq. 9.44 and Table 9.1. 1

CHAPTER 9. RELAXATION METHODS

174

Given our assumption that the component of the error associated with each eigenvector is equally likely to be excited, the asymptotic convergence rate is determined by the eigenvalue m of G ( I + H ; A) having maximum absolute value. Thus 1

Convergence rate  jmjmax ; m = 1; 2;


; M

(9.50)

In this section, we use the ODE analysis to nd the convergence rates of the three classical methods represented by Eqs. 9.30, 9.31, and 9.32. It is also instructive to inspect the eigenvectors and eigenvalues in the H ; A matrix for the three methods. This amounts to solving the generalized eigenvalue problem A~xm = m H~xm (9.51) 1

for the special case

B (1; ;2; 1)~xm = m B (; ; !2 ; 0)~xm

(9.52)

The generalized eigensystem for simple tridigonals is given in Appendix B.2. The three special cases considered below are obtained with a = 1, b = ;2, c = 1, d = ; , e = 2=!, and f = 0. To illustrate the behavior, we take M = 5 for the matrix order. This special case makes the general result quite clear.

9.4.1 The Point-Jacobi System

If = 0 and ! = 1 in Eq. 9.44, the ODE matrix H ; A reduces to simply B ( 12 ; ;1; 21 ). The eigensystem can be determined from Appendix B.1 since both d and f are zero. The eigenvalues are given by the equation  m  m = ;1 + cos M + 1 ; m = 1; 2; : : : ; M (9.53) The - relation for the explicit Euler method is m = 1 + m h. This relation can be plotted for any h. The plot for h = 1, the optimum stationary case, is shown in Fig. 9.1. For h < 1, the maximum jmj is obtained with m = 1, Fig. 9.2 and for h > 1, the maximum jm j is obtained with m = M , Fig. 9.3. Note that for h > 1:0 (depending on M ) there is the possibility of instability, i.e. jm j > 1:0. To obtain the optimal scheme we wish to minimize the maximum jm j which occurs when h = 1, j j = jM j and the best possible convergence rate is achieved:   jm jmax = cos M + 1 (9.54) 1

1

9.4. EIGENSYSTEMS OF THE CLASSICAL METHODS

175

For M = 40, we obtain jmjmax = 0:9971. Thus after 500 iterations the error content associated with each eigenvector is reduced to no more than 0.23 times its initial level. Again from Appendix B.1, the eigenvectors of H ; A are given by   ~xj = (xj )m = sin j m ; j = 1; 2; : : : ; M (9.55) M +1 This is a very \well-behaved" eigensystem with linearly independent eigenvectors and distinct eigenvalues. The rst 5 eigenvectors are simple sine waves. For M = 5, the eigenvectors can be written as 1

2 1=2 3 2 3 2 p3=2 3 p p 66 3=2 77 66 10 77 66 3=2 77 ~x = 666 p 1 777 ; ~x = 666 p0 777 ; ~x = 666 ;1 777 ; 64 3=2 75 64 0 75 64 ; 3=2 75 p 1 1=2 ; 3=2 3 2 p 3 2 3=2 1=2 p p 66 ; 3=2 77 66 ; 3=2 77 6 7 ~x = 66 p 0 77 ; ~x = 666 p1 777 64 3=2 75 64 ; 3=2 75 p 1=2 ; 3=2 1

3

2

4

5

(9.56)

The corresponding eigenvalues are, from Eq. 9.53

p

 = ;1 + 23 = ;0:134


 = ;1 + 12 = ;0:5  = ;1 = ;1:0 1

2

(9.57)

3

 = ;1 ; 12 = ;1:5 4

p

 = ;1 ; 23 = ;1:866


From Eq. 9.39, the numerical solution written in full is p ~n ; ~1 = c [1 ; (1 ; 3 )h]n~x 2 1 + c [1 ; (1 ; 2 )h]n~x 5

1

2

1

2

CHAPTER 9. RELAXATION METHODS

176 1.0

h = 1.0 m=1

m=5

Values are equal

σm m=4

m=2

Μ=5 m=3

0.0 -2.0

0.0

-1.0

Figure 9.1: The ;  relation for Point-Jacobi, h = 1; M = 5. + c [1 ; (1 3

)h]n~x

3

+ c [1 ; (1 + 12 )h]n~x p + c [1 ; (1 + 23 )h]n~x 4

4

5

5

(9.58)

9.4.2 The Gauss-Seidel System If and ! are equal to 1 in Eq. 9.44, the matrix eigensystem evolves from the relation

B (1; ;2; 1)~xm = mB (;1; 2; 0)~xm

(9.59)

which can be studied using the results in Appendix B.2. One can show that the H ; A matrix for the Gauss-Seidel method, AGS , is 1

AGS  B ; (;1; 2; 0)B (1; ;2; 1) = 1

9.4. EIGENSYSTEMS OF THE CLASSICAL METHODS Approaches 1.0

1.0

h

h < 1.0

De

ng

cr

si

ea

ea

si

cr

ng

De

σm

h

0.0 -2.0

0.0

-1.0

λ mh

Figure 9.2: The ;  relation for Point-Jacobi, h = 0:9; M = 5. Exceeds 1.0 at h

∼ ∼

1.072 / Ustable h > 1.072

1.0

h > 1.0

In

cr

h

ea

si

ng

h

ng si ea cr In

σm

M = 5 0.0 -2.0

-1.0

0.0

λmh

Figure 9.3: The ;  relation for Point-Jacobi, h = 1:1; M = 5.

177

CHAPTER 9. RELAXATION METHODS 2 ;1 1=2 3 1 66 0 ;3=4 1=2 77 2 66 0 1=8 ;3=4 1=2 77 3 66 77 66 0 1=16 1=8 ;3=4 1=2 77 4 (9.60) 66 0 1=32 1=16 1=8 ;3=4 1=2 77 5 ... 64 ... . . . . . . . . . . . . . . . 75 ... 0 1=2M


M The eigenvector structure of the Gauss-Seidel ODE matrix is quite interesting. If M is odd there are (M + 1)=2 distinct eigenvalues with corresponding linearly independent eigenvectors, and there are (M ; 1)=2 defective eigenvalues with corresponding 178

principal vectors. The equation for the nondefective eigenvalues in the ODE matrix is (for odd M ) M +1 ) ; m = 1 ; 2 ; : : : ; (9.61) m = ;1 + cos ( Mm +1 2 and the corresponding eigenvectors are given by   j;   m  M +1 ~xm = cos m sin j ; m = 1 ; 2 ; : : : ; (9.62) M +1 M +1 2 The - relation for h = 1, the optimum stationary case, is shown in Fig. 9.4. The m with the largest amplitude is obtained with m = 1. Hence the convergence rate is    (9.63) jm jmax = cos M + 1 Since this is the square of that obtained for the Point-Jacobi method, the error associated with the \worst" eigenvector is removed in half as many iterations. For M = 40, jm jmax = 0:9942. 250 iterations are required to reduce the error component of the worst eigenvector by a factor of roughly 0.23. The eigenvectors are quite unlike the Point-Jacobi set. They are no longer symmetrical, producing waves that are higher in amplitude on one side (the updated side) than they are on the other. Furthermore, they do not represent a common family for dierent values of M . The Jordan canonical form for M = 5 is 2

1

2

2 h~ i 66  h~ i  66 ; 6 X AGS X = JGS = 6 66 4 1

1

2

3 77 2~ 3 777 66  ~1 77 777  1 55 4 ~ 3

3

3

(9.64)

9.4. EIGENSYSTEMS OF THE CLASSICAL METHODS 1.0

179

h = 1.0

σm Μ=5

2 defective λ m 0.0 -2.0

0.0

-1.0

λmh

Figure 9.4: The ;  relation for Gauss-Seidel, h = 1:0; M = 5. The eigenvectors and principal vectors are all real. For M = 5 they can be written

2 3 2 p 3 2 3 2 3 2 3 3 = 2 1 = 2 1 0 66 3=4 77 66 p3=4 77 66 0 77 66 2 77 66 00 77 ~x = 666 3=4 777 ; ~x = 666 p0 777 ; ~x = 666 0 777 ; ~x = 666 ;1 777 ; ~x = 666 4 777 (9.65) 64 9=16 75 64 ; 3=16 75 64 0 75 64 0 75 64 ;4 75 p 9=32 0 0 1 ; 3=32 1

2

3

4

5

The corresponding eigenvalues are

 = ;1=4  = ;3=4 ) = ;1 (4) Defective, linked to (5)  Jordan block The numerical solution written in full is thus ~n ; ~1 = c (1 ; h )n~x 4 3 + c (1 ; 4h )n~x 1 2 3

3

1

2

1

2

(9.66)

CHAPTER 9. RELAXATION METHODS " # n n ( n ; 1) n n ; n ; + c (1 ; h) + c h 1! (1 ; h) + c h 2! (1 ; h) ~x   + c (1 ; h)n + c h n (1 ; h)n; ~x

180

3

4

4

5

+ c (1 ; h)n~x 5

1

5

1

1!

2

2

3

4

(9.67)

5

9.4.3 The SOR System

If = 1 and 2=! = x in Eq. 9.44, the ODE matrix is B ; (;1; x; 0)B (1; ;2; 1). One can show that this can be written in the form given below for M = 5. The generalization to any M is fairly clear. The H ; A matrix for the SOR method, ASOR  B ; (;1; x; 0)B (1; ;2; 1), is 1

1

1

2 x 0 0 66 ;2;x2x+ x x 0 1 66 ;2x + x x x; 2;x2x+ x x ; 2 x x x 664 ;2x + x x ; 2x + x x ; 2x + x x ; 2x ;2 + x 1 ; 2x + x x ; 2x + x x ; 2x + x x 4

5

4

3

4

2

3

2

3

2

4

4

3

2

4

3

3

4

2

3

2

2

4

4

3

3

2

4

3

4

0 0 0 x

4

3

; 2x

4

3 77 77 77 (9.68) 5

Eigenvalues of the system are given by

m = ;1 +

 !pm + zm  2

2

; m = 1; 2; : : : M

(9.69)

where

zm = [4(1 ; !) + ! pm] = pm = cos[m=(M + 1)] 2 2

1 2

If ! = 1, the system is Gauss-Seidel. If 4(1 ; !) + ! pm < 0; zm and m are complex. If ! is chosen such that 4(1 ; !) + ! p = 0; ! is optimum for the stationary case, and the following conditions hold: 2

2 2 1

1. Two eigenvalues are real, equal and defective. 2. If M is even, the remaining eigenvalues are complex and occur in conjugate pairs. 3. If M is odd, one of the remaining eigenvalues is real and the others are complex occurring in conjugate pairs.

9.4. EIGENSYSTEMS OF THE CLASSICAL METHODS

181

One can easily show that the optimum ! for the stationary case is    !opt = 2= 1 + sin M + 1 and for ! = !opt m = m ; 1    ~xm = mj; sin j m M +1

(9.70)

2

(9.71)

1

where

  q ! opt m = 2 pm + i p ; pm Using the explicit Euler method to integrate the ODE's, m = 1 ; h + hm , and if h = 1, the optimum value for the stationary case, the - relation reduces to that shown in Fig. 9.5. This illustrates the fact that for optimum stationary SOR all the jm j are identical and equal to !opt ; 1. Hence the convergence rate is jm jmax = !opt ; 1  (9.72)  !opt = 2= 1 + sin M + 1 2 1

2

2

For M = 40, jmjmax = 0:8578. Hence the worst error component is reduced to less than 0.23 times its initial value in only 10 iterations, much faster than both GaussSeidel and Point-Jacobi. In practical applications, the optimum value of ! may have to be determined by trial and error, and the bene t may not be as great. For odd M , there are two real eigenvectors and one real principal vector. The remaining linearly independent eigenvectors are all complex. For M = 5 they can be written 2 3 2 3 2 p3(1 3 2 3 )=2 1 = 2 ; 6 1 p p 66 1=2 77 66 9 77 66 3(1  i 2)=6 77 66 0 77 6 7 6 7 6 7 ~x = 66 1=3 77 ; ~x = 66 16 77 ; ~x ; = 66 p 77 ; ~x = 666 1=3 777 (9.73) 0p 64 1=6 75 64 13 75 64 3(5  i 2)=54 75 64 0 75 p p 1=18 6 1=9 3(7  4i 2)=162 The corresponding eigenvalues are  = ;2=3 (2) Defectiveplinked to   = ;(10 ; 2p2i)=9  = ;(10 + 2 2i)=9  = ;4=3 (9.74) 1

2

34

5

1

1

3

4

5

CHAPTER 9. RELAXATION METHODS

182 1.0

h = 1.0 2 real defective λ m

σm

2 complex λ m

1 real λ m

Μ=5 0.0 -2.0

0.0

-1.0

λmh

Figure 9.5: The ;  relation for optimum stationary SOR, M = 5, h = 1. The numerical solution written in full is

~n ; ~1 = + + + +

[c (1 ; 2h=3)n + c nh(1 ; 2h=3)n; ]~x c (1 ; 2h=3)n~xp c [1 ; (10 ; 2p2i)h=9]n~x c [1 ; (10 + 2 2i)h=9]n~x c (1 ; 4h=3)n~x 1

2

1

2

2

3

3

4

4

5

1

5

(9.75)

9.5 Nonstationary Processes In classical terminology a method is said to be nonstationary if the conditioning matrices, H and C , are varied at each time step. This does not change the steadystate solution A;b ~f b , but it can greatly aect the convergence rate. In our ODE approach this could also be considered and would lead to a study of equations with nonconstant coecients. It is much simpler, however, to study the case of xed H and C but variable step size, h. This process changes the Point-Jacobi method to Richardson's method in standard terminology. For the Gauss-Seidel and SOR methods it leads to processes that can be superior to the stationary methods. The nonstationary form of Eq. 9.39 is 1

9.5. NONSTATIONARY PROCESSES

183

N N ~N = c ~x Y (1 +  hn ) +


+ cm~xm Y (1 + mhn ) 1

1

n=1

+


+ cM ~xM

1

N Y n=1

n=1

(1 + M hn ) + ~1

(9.76)

where the symbol  stands for product. Since hn can now be changed at each step, the error term can theoretically be completely eliminated in M steps by taking hm = ;1=m , for m = 1; 2;


; M . However, the eigenvalues m are generally unknown and costly to compute. It is therefore unnecessary and impractical to set hm = ;1=m for m = 1; 2; : : : ; M . We will see that a few well chosen h's can reduce whole clusters of eigenvectors associated with nearby 's in the m spectrum. This leads to the concept of selectively annihilating clusters of eigenvectors from the error terms as part of a total iteration process. This is the basis for the multigrid methods discussed in Chapter 10. Let us consider the very important case when all of the m are real and negative (remember that they arise from a conditioned matrix so this constraint is not unrealistic for quite practical cases). Consider one of the error terms taken from M N ~eN  ~N ; ~1 = X cm~xm Y (1 + mhn) m=1

n=1

(9.77)

and write it in the form

cm~xm Pe(m )  cm~xm

N Y n=1

(1 + m hn)

(9.78)

where Pe signi es an \Euler" polynomial. Now focus attention on the polynomial (Pe)N () = (1 + h )(1 + h )


(1 + hN ) 1

2

(9.79)

treating it as a continuous function of the independent variable . In the annihilation process mentioned after Eq. 9.76, we considered making the error exactly zero by taking advantage of some knowledge about the discrete values of m for a particular case. Now we pose a less demanding problem. Let us choose the hn so that the maximum value of (Pe)N () is as small as possible for all  lying between a and b such that b    a  0. Mathematically stated, we seek max j(Pe)N ()j = minimum ; with(Pe)N (0) = 1

b a

(9.80)

CHAPTER 9. RELAXATION METHODS

184

This problem has a well known solution due to Markov. It is

TN 2 ; ;a ; b a b ! (Pe)N () = TN ;a;;b a b

!

(9.81)

where

TN (y) = cos(N arccos y)

(9.82)

are the Chebyshev polynomials along the interval ;1  y  1 and

 q N 1  q N 1 TN (y) = 2 y + y ; 1 + 2 y ; y ; 1 (9.83) are the Chebyshev polynomials for jyj > 1. In relaxation terminology this is generally referred to as Richardson's method, and it leads to the nonstationary step size choice given by 2

(

2

"

#)

1 = 1 ; ;  + ( ;  ) cos (2n ; 1) ; n = 1; 2; : : : N b a hn 2 b a 2N

(9.84)

Remember that all  are negative real numbers representing the magnitudes of m in an eigenvalue spectrum. The error in the relaxation process represented by Eq. 9.76 is expressed in terms Q ~ of a set of eigenvectors, xm , ampli ed by the coecients cm (1 + m hn). With each eigenvector there is a corresponding eigenvalue. Eq. 9.84 gives us the best choice of a series of hn that will minimize the amplitude of the error carried in the eigenvectors associated with the eigenvalues between b and a . As an example for the use of Eq. 9.84, let us consider the following problem: Minimize the maximum error associated with the  eigenvalues in the interval ;2    ;1 using only 3 iterations. The three values of h which satisfy this problem are

"

hn = 2= 3 ; cos (2n ;6 1)

#!

(9.85)

(9.86)

9.5. NONSTATIONARY PROCESSES

185

and the amplitude of the eigenvector is reduced to (Pe) () = T (2 + 3)=T (3)

(9.87)

n p p o T (3) = [3 + 8] + [3 ; 8] =2  99

(9.88)

3

where

3

3

3

3

3

A plot of Eq. 9.87 is given in Fig. 9.6 and we see that the amplitudes of all the eigenvectors associated with the eigenvalues in the range ;2    ;1 have been reduced to less than about 1% of their initial values. The values of h used in Fig. 9.6 are

p

h = 4=(6 ; 3) h = 4=(6 ; p 0) h = 4=(6 + 3) 1

2 3

Return now to Eq. 9.76. This was derived from Eq. 9.37 on the condition that the explicit Euler method, Eq. 9.41, was used to integrate the basic ODE's. If instead the implicit trapezoidal rule

n = n + 21 h(0n + 0n)

(9.89)

0 1 1 M N 1 + hn m C ~N = X cm~xm Y B CA + ~1 B@ 2 1 m n 1 ; 2 hnm

(9.90)

+1

+1

is used, the nonstationary formula

=1

=1

would result. This calls for a study of the rational \trapezoidal" polynomial, Pt: 0 1 1 N Y B 1 + 2 hn  CC (Pt )N () = B (9.91) @ 1 A n 1 ; 2 hn under the same constraints as before, namely that =1

max j(Pt)N ()j = minimum , with (Pt)N (0) = 1

b a

(9.92)

CHAPTER 9. RELAXATION METHODS

186

1 0.9 0.8 0.7

(Pe )3 (λ)

0.6 0.5 0.4 0.3 0.2 0.1 0 −2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

−1

−0.8

−0.6

−0.4

−0.2

0

λ

0.02

0.015

0.01

e 3

(P ) (λ)

0.005

0

−0.005

−0.01

−0.015

−0.02 −2

−1.8

−1.6

−1.4

−1.2

λ

Figure 9.6: Richardson's method for 3 steps, minimization over ;2    ;1.

9.6. PROBLEMS

187

The optimum values of h can also be found for this problem, but we settle here for the approximation suggested by Wachspress

!

2 = ; a n; = N ; ; n = 1; 2;


; N (9.93) b hn b This process is also applied to problem 9.85. The results for (Pt) () are shown in Fig. 9.7. The error amplitude is about 1/5 of that found for (Pe) () in the same interval of . The values of h used in Fig. 9.7 are h = 1p h = 2 h = 2 (

1) (

1)

3

3

1 2

3

9.6 Problems 1. Given a relaxation method in the form H
~n = A~n ; f~ show that

~n = Gn~ + (I ; Gn)A; f~ 1

0

where G = I + H ; A. 2. For a linear system of the form (A + A )x = b, consider the iterative method 1

1

2

(I + A )~x = (I ; A )xn + b (I + A )xn = (I ; A )~x + b where  is a parameter. Show that this iterative method can be written in the form H (xk ; xk ) = (A + A )xk ; b 1

2

+1

2

+1

1

1

2

Determine the iteration matrix G if  = ;1=2. 3. Using Appendix B.2, nd the eigenvalues of H ; A for the SOR method with A = B (4 : 1; ;2; 1) and ! = !opt. (You do not have to nd H ; . Recall that the eigenvalues of H ; A satisfy A~xm = mH~xm .) Find the numerical values, not just the expressions. Then nd the corresponding jm j values. 1

1

1

CHAPTER 9. RELAXATION METHODS

188

1 0.9 0.8 0.7

t 3

(P ) (λ)

0.6 0.5 0.4 0.3 0.2 0.1 0 −2

−1.8

−1.6

−1.4

−1.2

−1

λ

−0.8

−0.6

−0.4

−0.2

0

−0.8

−0.6

−0.4

−0.2

0

−3

2

x 10

1.5

1

(Pt )3 (λ)

0.5

0

−0.5

−1

−1.5

−2 −2

−1.8

−1.6

−1.4

−1.2

−1

λ

Figure 9.7: Wachspress method for 3 steps, minimization over ;2    ;1.

9.6. PROBLEMS

189

4. Solve the following equation on the domain 0  x  1 with boundary conditions u(0) = 0, u(1) = 1:

@ u ; 6x = 0 @x For the initial condition, use u(x) = 0. Use second-order centered dierences on a grid with 40 cells (M = 39). Iterate to steady state using (a) the point-Jacobi method, (b) the Gauss-Seidel method, (c) the SOR method with the optimum value of !, and (d) the 3-step Richardson method derived in Section 9.5. Plot the solution after the residual is reduced by 2, 3, and 4 orders of magnitude. Plot the logarithm of the L -norm of the residual vs. the number of iterations. Determine the asymptotic convergence rate. Compare with the theoretical asymptotic convergence rate. 2

2

2

190

CHAPTER 9. RELAXATION METHODS

Chapter 10 MULTIGRID The idea of systematically using sets of coarser grids to accelerate the convergence of iterative schemes that arise from the numerical solution to partial dierential equations was made popular by the work of Brandt. There are many variations of the process and many viewpoints of the underlying theory. The viewpoint presented here is a natural extension of the concepts discussed in Chapter 9.

10.1 Motivation 10.1.1 Eigenvector and Eigenvalue Identi cation with Space Frequencies Consider the eigensystem of the model matrix B (1; ;2; 1). The eigenvalues and

eigenvectors are given in Sections 4.3.2 and 4.3.3, respectively. Notice that as the magnitudes of the eigenvalues increase, the space-frequency (or wavenumber) of the corresponding eigenvectors also increase. That is, if the eigenvalues are ordered such that

j j  j j 


 jM j 1

(10.1)

2

then the corresponding eigenvectors are ordered from low to high space frequencies. This has a rational explanation from the origin of the banded matrix. Note that @ sin(mx) = ;m sin(mx) (10.2) @x and recall that   (10.3) xx~ =
1x B (1; ;2; 1)~ = X
1x D(~) X ; ~ 191 2

2

2

2

2

1

CHAPTER 10. MULTIGRID

192

where D(~) is a diagonal matrix containing the eigenvalues. We have seen that X ; ~ represents a sine transform, and X ~, a sine synthesis. Therefore, the operation
1x D(~) represents the numerical approximation of the multiplication of the appropriate sine wave by the negative square of its wavenumber, ;m . One nds that 1  =  M + 1  ;2 + 2 cos  m   ;m ; m << M (10.4)  M +1
x m Hence, the correlation of large magnitudes of m with high space-frequencies is to be expected for these particular matrix operators. This is consistent with the physics of diusion as well. However, this correlation is not necessary in general. In fact, the complete counterexample of the above association is contained in the eigensystem for B ( 21 ; 1; 12 ). For this matrix one nds, from Appendix B, exactly the opposite behavior. 1

2

2

2

2

2

10.1.2 Properties of the Iterative Method The second key motivation for multigrid is the following:

 Many iterative methods reduce error components corresponding to eigenvalues of large amplitude more eectively than those corresponding to eigenvalues of small amplitude.

This is to be expected of an iterative method which is time accurate. It is also true, for example, of the Gauss-Seidel method and, by design, of the Richardson method described in Section 9.5. The classical point-Jacobi method does not share this property. As we saw in Section 9.4.1, this method produces the same value of jj for min and max . However, the property can be restored by using h < 1, as shown in Fig. 9.2. When an iterative method with this property is applied to a matrix with the above correlation between the modulus of the eigenvalues and the space frequency of the eigenvectors, error components corresponding to high space frequencies will be reduced more quickly than those corresponding to low space frequencies. This is the key concept underlying the multigrid process.

10.2 The Basic Process First of all we assume that the dierence equations representing the basic partial dierential equations are in a form that can be related to a matrix which has certain

10.2. THE BASIC PROCESS

193

basic properties. This form can be arrived at \naturally" by simply replacing the derivatives in the PDE with dierence schemes, as in the example given by Eq. 3.27, or it can be \contrived" by further conditioning, as in the examples given by Eq. 9.11. The basic assumptions required for our description of the multigrid process are: 1. The problem is linear. 2. The eigenvalues, m , of the matrix are all real and negative. 3. The m are fairly evenly distributed between their maximum and minimum values. 4. The eigenvectors associated with the eigenvalues having largest magnitudes can be correlated with high frequencies on the dierencing mesh. 5. The iterative procedure used greatly reduces the amplitudes of the eigenvectors associated with eigenvalues in the range between 21 jjmax and jjmax. These conditions are sucient to ensure the validity of the process described next. Having preconditioned (if necessary) the basic nite dierencing scheme by a procedure equivalent to the multiplication by a matrix C , we are led to the starting formulation C [Ab~1 ; ~f b ] = 0 (10.5) where the matrix formed by the product CAb has the properties given above. In Eq. 10.5, the vector ~f b represents the boundary conditions and the forcing function, if any, and ~1 is a vector representing the desired exact solution. We start with some initial guess for ~1 and proceed through n iterations making use of some iterative process that satis es property 5 above. We do not attempt to develop an optimum procedure here, but for clarity we suppose that the three-step Richardson method illustrated in Fig. 9.6 is used. At the end of the three steps we nd ~r, the residual, where ~r = C [Ab~ ; ~f b] (10.6) Recall that the ~ used to compute ~r is composed of the exact solution ~1 and the error ~e in such a way that A~e ; ~r = 0 (10.7) where A  CAb (10.8)

CHAPTER 10. MULTIGRID

194 If one could solve Eq. 10.7 for ~e then ~1 = ~ ; ~e

(10.9)

Thus our goal now is to solve for ~e. We can write the exact solution for ~e in terms of the eigenvectors of A, and the  eigenvalues of the Richardson process in the form: M= M ~e = X cm~xm Y [(m hn)] + X cm~xm Y [(mhn)] 2

3

m=1

3

n=1

n=1

m=M=2+1

|

{z

very low amplitude

(10.10)

}

Combining our basic assumptions, we can be sure that the high frequency content of ~e has been greatly reduced (about 1% or less of its original value in the initial guess). In addition, assumption 4 ensures that the error has been smoothed. Next we construct a permutation matrix which separates a vector into two parts, one containing the odd entries, and the other the even entries of the original vector (or any other appropriate sorting which is consistent with the interpolation approximation to be discussed below). For a 7-point example 2 66 66 66 66 66 64

e e e e e e e

2 4 6 1 3 5

3 2 77 66 77 66 77 66 77 = 66 77 66 75 64

7

0 0 0 1 0 0 0

1 0 0 0 0 0 0

0 0 0 0 1 0 0

0 1 0 0 0 0 0

0 0 0 0 0 1 0

32

0 0 1 0 0 0 0

0 76 0 77 66 0 77 66 0 777 666 0 77 66 0 75 64 1

e e e e e e e

1 2 3 4 5 6

3 77 77 " # 77 ~ 77 ; ~ee = P ~e eo 77 75

(10.11)

7

Multiply Eq. 10.7 from the left by P and, since a permutation matrix has an inverse which is its transpose, we can write PA[P ; P ]~e = P ~r (10.12) 1

The operation PAP ; partitions the A matrix to form 1

2 3 A1 A2 " ~e # " ~r # 64 75 e = e ~ ~ro A 3 A4 e o

(10.13)

A ~ee + A ~eo = ~re

(10.14)

Notice that 1

2

10.2. THE BASIC PROCESS

195

is an exact expression. At this point we make our one crucial assumption. It is that there is some connection between ~ee and ~eo brought about by the smoothing property of the Richardson relaxation procedure. Since the top half of the frequency spectrum has been removed, it is reasonable to suppose that the odd points are the average of the even points. For example e  21 (ea + e ) e  12 (e + e ) or ~eo = A0 ~ee (10.15) e  12 (e + e ) e  12 (e + eb ) 1

2

3

2

4

5

4

6

7

6

2

It is important to notice that ea and eb represent errors on the boundaries where the error is zero if the boundary conditions are given. It is also important to notice that we are dealing with the relation between ~e and ~r so the original boundary conditions and forcing function (which are contained in ~f in the basic formulation) no longer appear in the problem. Hence, no aliasing of these functions can occur in subsequent steps. Finally, notice that, in this formulation, the averaging of ~e is our only approximation, no operations on ~r are required or justi ed. If the boundary conditions are Dirichlet, ea and eb are zero, and one can write for the example case 2 1 0 0 6 1 A02 = 2 664 10 11 01

0 0 1

3 77 75

With this approximation Eq. 10.14 reduces to A ~ee + A A0 ~ee = ~re 1

2

2

(10.16)

(10.17)

or

Ac~ee ; ~re = 0

(10.18)

where

Ac = [A + A A0 ] 1

2

2

(10.19)

CHAPTER 10. MULTIGRID

196

The form of Ac, the matrix on the coarse mesh, is completely determined by the choice of the permutation matrix and the interpolation approximation. If the original A had been B (7 : 1; ;2; 1), our 7-point example would produce 2 ;2 66 66 66 ; 1 PAP = 666 1 66 66 1 4

;2 1 1

1

;2 ;2 1 1

1 1

;2

1 1

;2

3 77 1 777 2 A A 3 77 6 1 2 7 77 = 4 5 77 A3 A4 77 5

(10.20)

;2

and Eq. 10.18 gives A

0

2 z z2 }| {3 2z }| {3 2 1 }| ;2 1 1 16 1 1 64 7 6 ;2 5 + 4 1 1 75
664 1 1 ;2 1 1 2 1

A2

A1

Ac 3{ 2z }| ; 1 1 =2 77 6 75 = 4 1=2 ;1 1=2

1=2 ;1

3{ 75 (10.21)

If the boundary conditions are mixed Dirichlet-Neumann, A in the 1-D model equation is B (1; ~b; 1) where ~b = [;2; ;2; :::; ;2; ;1]T . The eigensystem is given by Eq. B.19. It is easy to show that the high space-frequencies still correspond to the eigenvalues with high magnitudes, and, in fact, all of the properties given in Section 10.1 are met. However, the eigenvector structure is dierent from that given in Eq. 9.55 for Dirichlet conditions. In the present case they are given by " !# (2 m ; 1)  xjm = sin j 2M + 1 ; m = 1; 2;


; M (10.22) and are illustrated in Fig. 10.1. All of them go through zero on the left (Dirichlet) side, and all of them re ect on the right (Neumann) side. For Neumann conditions, the interpolation formula in Eq. 10.15 must be changed. In the particular case illustrated in Fig. 10.1, eb is equal to eM . If Neumann conditions are on the left, ea = e . When eb = eM , the example in Eq. 10.16 changes to 2 1 0 03 6 7 A0 = 21 664 10 11 01 775 (10.23) 0 0 2 1

2

10.2. THE BASIC PROCESS

197

X

Figure 10.1: Eigenvectors for the mixed Dirichlet{Neumann case. The permutation matrix remains the same and both A and A in the partitioned matrix PAP ; are unchanged (only A is modi ed by putting ;1 in the lower right element). Therefore, we can construct the coarse matrix from 1

1

A

0

2 z }| {3 z2 }| {3 2 1 }| z2 ;2 1 1 16 1 1 64 7 6 ;2 5 + 4 1 1 75
664 1 1 ;2 1 1 2

A1

2

4

A2

2

Ac 3{ 2z }| ; 1 1 = 2 77 6 75 = 4 1=2 ;1

3{ 1=2 75 (10.24)

1=2 ;1=2

which gives us what we might have \expected." We will continue with Dirichlet boundary conditions for the remainder of this Section. At this stage, we have reduced the problem from B (1; ;2; 1)~e = ~r on the ne mesh to 21 B (1; ;2; 1)~ee = ~re on the next coarser mesh. Recall that our goal is to solve for ~e, which will provide us with the solution ~1 using Eq. 10.9. Given ~ee computed on the coarse grid (possibly using even coarser grids), we can compute ~eo using Eq. 10.15, and thus ~e. In order to complete the process, we must now determine the relationship between ~ee and ~e. In order to examine this relationship, we need to consider the eigensystems of A and Ac: A = X X ; ; Ac = XccXc; (10.25) For A = B (M : 1; ;2; 1) the eigenvalues and eigenvectors are   m    m  j = 1; 2;


; M ~ m = ;2 1 ; cos M + 1 ; xm = sin j M + 1 ; m = 1; 2;


; M (10.26) 1

1

CHAPTER 10. MULTIGRID

198

Based on our assumptions, the most dicult error mode to eliminate is that with m = 1, corresponding to        = ;2 1 ; cos M + 1 ; ~x = sin j M + 1 ; j = 1; 2;


; M (10.27) For example, with M = 51,  = ;0:003649. If we restrict our attention to odd M , then Mc = (M ; 1)=2 is the size of Ac. The eigenvalue and eigenvector corresponding to m = 1 for the matrix Ac = 12 B (Mc; 1; ;2; 1) are        ~xc) = sin j 2 (c) = ; 1 ; cos M2+ ; ( 1 M + 1 ; j = 1; 2;


; Mc (10.28) For M = 51 (Mc = 25), we obtain (c) = ;0:007291 = 1:998 . As M increases, (c) approaches 2 . In addition, one can easily see that (~xc) coincides with ~x at every second point of the latter vector, that is, it contains the even elements of ~x . Now let us consider the case in which all of the error consists of the eigenvector component ~x , i.e., ~e = ~x . Then the residual is ~r = A~x =  ~x (10.29) 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

and the residual on the coarse grid is ~re =  (~xc) 1

(10.30)

1

since (~xc) contains the even elements of ~x . The exact solution on the coarse grid satis es ~ee = A;c ~re = Xc;c Xc;  (~xc) (10.31) 2 3 1 66 0 77 =  Xc;c 66 .. 77 4 . 5 0 (10.32) 2 3 1=(c) 66 0 77 =  Xc 66 .. 77 4 . 5 0 (10.33) 1

1

1

1

1

1

1

1

1

1

1

10.2. THE BASIC PROCESS

199 =

 (~x ) (c) c 1

1

1

 12 (~xc)

1

(10.34) (10.35)

Since our goal is to compute ~e = ~x , in addition to interpolating ~ee to the ne grid (using Eq. 10.15), we must multiply the result by 2. This is equivalent to solving 1

1 A ~e = ~r 2 ce e

or

(10.36)

1 B (M : 1; ;2; 1)~e = ~r (10.37) c e e 4 In our case, the matrix A = B (M : 1; ;2; 1) comes from a discretization of the diusion equation, which gives Ab =
x B (M : 1; ;2; 1) (10.38) and the preconditioning matrix C is simply 2

C =
x I 2

(10.39)

Applying the discretization on the coarse grid with the same preconditioning matrix as used on the ne grid gives, since
xc = 2
x,
x B (M : 1; ;2; 1) = 1 B (M : 1; ;2; 1) C
x B (Mc : 1; ;2; 1) =
c c x 4 2

2

c

2

c

(10.40)

which is precisely the matrix appearing in Eq. 10.37. Thus we see that the process is recursive. The problem to be solved on the coarse grid is the same as that solved on the ne grid. The remaining steps required to complete an entire multigrid process are relatively straightforward, but they vary depending on the problem and the user. The reduction can be, and usually is, carried to even coarser grids before returning to the nest level. However, in each case the appropriate permutation matrix and the interpolation approximation de ne both the down- and up-going paths. The details of nding optimum techniques are, obviously, quite important but they are not discussed here.

CHAPTER 10. MULTIGRID

200

10.3 A Two-Grid Process

We now describe a two-grid process for the linear problem A~ = f~, which can be easily generalized to a process with an arbitrary number of grids due to the recursive nature of multigrid. Extension to nonlinear problems requires that both the solution and the residual be transferred to the coarse grid in a process known as full approximation storage multigrid. 1. Perform n iterations of the selected relaxation method on the ne grid, starting with ~ = ~n. Call the result ~ . This gives ~ = Gn1 ~n + (I ; Gn1 ) A; f~ (10.41) where G = I + H; A (10.42) and H is de ned as in Chapter 9 (e.g., Eq. 9.21). Next compute the residual based on ~ : ~r = A~ ; f~ = AGn1 ~n + A (I ; Gn1 ) A; f~ ; f~ = AGn1 ~n ; AGn1 A; f~ (10.43) 1

(1)

(1)

1

1

1

1

1

1

1

1

1 (1)

(1)

(1)

1

1

1

1

1

1

2. Transfer (or restrict) ~r to the coarse grid: ~r = R ~r (10.44) In our example in the preceding section, the restriction matrix is 2 3 0 1 0 0 0 0 0 R = 64 0 0 0 1 0 0 0 75 (10.45) 0 0 0 0 0 1 0 that is, the rst three rows of the permutation matrix P in Eq. 10.11. This type of restriction is known as \simple injection." Some form of weighted restriction can also be used. (1)

(2)

2 (1) 1

2 1

3. Solve the problem A ~e = ~r on the coarse grid exactly: ~e = A; ~r 2

(2)

(2)

2

(2)

1 2

2

1 (2)

(10.46)

See problem 1 of Chapter 9. Note that the coarse grid matrix denoted A2 here was denoted Ac in the preceding section.

10.3. A TWO-GRID PROCESS

201

Here A can be formed by applying the discretization on the coarse grid. In the preceding example (eq. 10.40), A = 41 B (Mc : 1; ;2; 1). It is at this stage that the generalization to a multigrid procedure with more than two grids occurs. If this is the coarsest grid in the sequence, solve exactly. Otherwise, apply the two-grid process recursively. 2

2

4. Transfer (or prolong) the error back to the ne grid and update the solution:

~n = ~ ; I ~e (1)

+1

In our example, the prolongation matrix is 2 66 66 6 I21 = 666 66 64

1=2 1 1=2 0 0 0 0

(10.47)

1 (2) 2

0 0 1=2 1 1=2 0 0

0 0 0 0 1=2 1 1=2

3 77 77 77 77 77 75

(10.48)

which follows from Eq. 10.15. Combining these steps, one obtains

~n = [I ; I A; R A]Gn1 ~ n ; [I ; I A; R A]Gn1 A; f~ + A; f~ 1 2

+1

2

1

2 1

1 2

1

2

1

2 1

1

1

(10.49)

1

Thus the basic iteration matrix is [I ; I A; R A]Gn1 1 2

2

1

2 1

(10.50)

1

The eigenvalues of this matrix determine the convergence rate of the two-grid process. The basic iteration matrix for a three-grid process is found from Eq. 10.50 by replacing A; with (I ; G )A; , where 2

1

2 3

2

1

G = [I ; I A; R A ]Gn2 2 3

2 3

3

1

3 2

2

(10.51)

2

In this expression n is the number of relaxation steps on grid 2, I and R are the transfer operators between grids 2 and 3, and A is obtained by discretizing on grid 3. Extension to four or more grids proceeds in similar fashion. 2 3

2

3

3 2

202

10.4 Problems

CHAPTER 10. MULTIGRID

1. Derive Eq. 10.51. 2. Repeat problem 4 of Chapter 9 using a four-grid multigrid method together with (a) the Gauss-Seidel method, (b) the 3-step Richardson method derived in Section 9.5. Solve exactly on the coarsest grid. Plot the solution after the residual is reduced by 2, 3, and 4 orders of magnitude. Plot the logarithm of the L -norm of the residual vs. the number of iterations. Determine the asymptotic convergence rate. Calculate the theoretical asymptotic convergence rate and compare. 2

Chapter 11 NUMERICAL DISSIPATION Up to this point, we have emphasized the second-order centered-dierence approximations to the spatial derivatives in our model equations. We have seen that a centered approximation to a rst derivative is nondissipative, i.e., the eigenvalues of the associated circulant matrix (with periodic boundary conditions) are pure imaginary. In processes governed by nonlinear equations, such as the Euler and Navier-Stokes equations, there can be a continual production of high-frequency components of the solution, leading, for example, to the production of shock waves. In a real physical problem, the production of high frequencies is eventually limited by viscosity. However, when we solve the Euler equations numerically, we have neglected viscous eects. Thus the numerical approximation must contain some inherent dissipation to limit the production of high-frequency modes. Although numerical approximations to the Navier-Stokes equations contain dissipation through the viscous terms, this can be insucient, especially at high Reynolds numbers, due to the limited grid resolution which is practical. Therefore, unless the relevant length scales are resolved, some form of added numerical dissipation is required in the numerical solution of the Navier-Stokes equations as well. Since the addition of numerical dissipation is tantamount to intentionally introducing nonphysical behavior, it must be carefully controlled such that the error introduced is not excessive. In this Chapter, we discuss some dierent ways of adding numerical dissipation to the spatial derivatives in the linear convection equation and hyperbolic systems of PDE's. 203

CHAPTER 11. NUMERICAL DISSIPATION

204

11.1 One-Sided First-Derivative Space Dierencing We investigate the properties of one-sided spatial dierence operators in the context of the biconvection model equation given by

@u = ;a @u @t @x

(11.1)

with periodic boundary conditions. Consider the following point operator for the spatial derivative term

;a [;(1 + )u + 2 u + (1 ; )u ] ;a(x u)j = 2
j ;1 j j +1 x = ;a [(;uj;1 + uj+1) + (;uj;1 + 2uj ; uj+1)] 2
x

(11.2)

The second form shown divides the operator into an antisymmetric component (;uj;1+ uj+1)=2
x and a symmetric component (;uj;1 + 2uj ; uj+1)=2
x. The antisymmetric component is the second-order centered dierence operator. With 6= 0, the operator is only rst-order accurate. A backward dierence operator is given by = 1 and a forward dierence operator is given by = ;1. For periodic boundary conditions the corresponding matrix operator is

;a B (;1 ; ; 2 ; 1 ; ) ;ax = 2
x p The eigenvalues of this matrix are

   2m   2m  ; a m =
x 1 ; cos M + i sin M for m = 0; 1; : : : ; M ; 1 If a is positive, the forward dierence operator ( = ;1) produces Re(m ) > 0, the centered dierence operator ( = 0) produces Re(m ) = 0, and the backward dierence operator produces Re(m) < 0. Hence the forward dierence operator is inherently unstable while the centered and backward operators are inherently stable. If a is negative, the roles are reversed. When Re(m) = 0, the solution will either grow or decay with time. In either case, our choice of dierencing scheme produces nonphysical behavior. We proceed next to show why this occurs.

11.2. THE MODIFIED PARTIAL DIFFERENTIAL EQUATION

205

11.2 The Modi ed Partial Dierential Equation

First carry out a Taylor series expansion of the terms in Eq. 11.2. We are lead to the expression

2

!

!

!

!

3

2 3 3 4 4 2 @ u +
x @ u ;
x @ u + : : :5 (xu)j = 2
1 x 42
x @u ;
x @x j @x2 j 3 @x3 j 12 @x4 j

We see that the antisymmetric portion of the operator introduces odd derivative terms in the truncation error while the symmetric portion introduces even derivatives. Substituting this into Eq. 11.1 gives

@u = ;a @u + a
x @ 2 u ; a
x2 @ 3 u + a
x3 @ 4 u + : : : (11.3) @t @x 2 @x2 6 @x3 24 @x4 This is the partial dierential equation we are really solving when we apply the approximation given by Eq. 11.2 to Eq. 11.1. Notice that Eq. 11.3 is consistent with Eq. 11.1, since the two equations are identical when
x ! 0. However, when we use a computer to nd a numerical solution of the problem,
x can be small but it is not zero. This means that each term in the expansion given by Eq. 11.3 is excited to some degree. We refer to Eq. 11.3 as the modi ed partial dierential equation. We proceed next to investigate the implications of this concept. Consider the simple linear partial dierential equation @u = ;a @u +  @ 2 u + @ 3 u +  @ 4 u (11.4) @t @x @x2 @x3 @x4 Choose periodic boundary conditions and impose an initial condition u = eix. Under these conditions there is a wave-like solution to Eq. 11.4 of the form u(x; t) = eixe(r+is)t provided r and s satisfy the condition r + is = ;ia ; 2 ; i 3 + 4 or r = ;2 ( ; 2 ); s = ;(a + 2 ) The solution is composed of both amplitude and phase terms. Thus u = e| ;2 ({z;2}) e| i[x;({za+ 2 )t}] (11.5) amplitude phase

206

CHAPTER 11. NUMERICAL DISSIPATION

It is important to notice that the amplitude of the solution depends only upon  and  , the coecients of the even derivatives in Eq. 11.4, and the phase depends only on a and , the coecients of the odd derivatives. If the wave speed a is positive, the choice of a backward dierence scheme ( = 1) produces a modi ed PDE with  ; 2 > 0 and hence the amplitude of the solution decays. This is tantamount to deliberately adding dissipation to the PDE. Under the same condition, the choice of a forward dierence scheme ( = ;1) is equivalent to deliberately adding a destabilizing term to the PDE. By examining the term governing the phase of the solution in Eq. 11.5, we see that the speed of propagation is a + 2 . Referring to the modi ed PDE, Eq. 11.3 we have = ;a
x2 =6. Therefore, the phase speed of the numerical solution is less than the actual phase speed. Furthermore, the numerical phase speed is dependent upon the wavenumber . This we refer to as dispersion. Our purpose here is to investigate the properties of one-sided spatial dierencing operators relative to centered dierence operators. We have seen that the threepoint centered dierence approximation of the spatial derivative produces a modi ed PDE that has no dissipation (or ampli cation). One can easily show, by using the antisymmetry of the matrix dierence operators, that the same is true for any centered dierence approximation of a rst derivative. As a corollary, any departure from antisymmetry in the matrix dierence operator must introduce dissipation (or ampli cation) into the modi ed PDE. Note that the use of one-sided dierencing schemes is not the only way to introduce dissipation. Any symmetric component in the spatial operator introduces dissipation (or ampli cation). Therefore, one could choose = 1=2 in Eq. 11.2. The resulting spatial operator is not one-sided but it is dissipative. Biased schemes use more information on one side of the node than the other. For example, a third-order backward-biased scheme is given by (xu)j = 6
1 x (uj;2 ; 6uj;1 + 3uj + 2uj+1) = 1 [(uj;2 ; 8uj;1 + 8uj+1 ; uj+2) 12
x + (uj;2 ; 4uj;1 + 6uj ; 4uj+1 + uj+2)]

(11.6)

The antisymmetric component of this operator is the fourth-order centered dierence operator. The symmetric component approximates
x3 uxxxx=12. Therefore, this operator produces fourth-order accuracy in phase with a third-order dissipative term.

11.3. THE LAX-WENDROFF METHOD

11.3 The Lax-Wendro Method

207

In order to introduce numerical dissipation using one-sided dierencing, backward dierencing must be used if the wave speed is positive, and forward dierencing must be used if the wave speed is negative. Next we consider a method which introduces dissipation independent of the sign of the wave speed, known as the Lax-Wendro method. This explicit method diers conceptually from the methods considered previously in which spatial dierencing and time-marching are treated separately. Consider the following Taylor-series expansion in time: 1 h2 @ 2 u + O(h3) + u(x; t + h) = u + h @u @t 2 @t2

(11.7)

First replace the time derivatives with space derivatives according to the PDE (in @u this case, the linear convection equation @u @t + a @x = 0). Thus

@ 2 u = a2 @ 2 u @u = ;a @u ; (11.8) @t @x @t2 @x2 Now replace the space derivatives with three-point centered dierence operators, giving !2 1 ah 1 ah ( n ) ( n ) ( n +1) ( n ) uj = uj ; 2
x (uj+1 ; uj;1) + 2
x (u(jn+1) ; 2u(jn) + u(jn;)1) (11.9) This is the Lax-Wendro method applied to the linear convection equation. It is a fully-discrete nite-dierence scheme. There is no intermediate semi-discrete stage. For periodic boundary conditions, the corresponding fully-discrete matrix operator is 0 2 !23 !2 2 !2 31 1 ah ah ah 1 ah ah ~un+1 = Bp @ 2 4
x +
x 5 ; 1 ;
x ; 2 4;
x +
x 5A ~un The eigenvalues of this matrix are

!2   2m  ah  2m  ah m = 1 ;
x 1 ; cos M ; i
x sin M for m = 0; 1; : : : ; M ; 1

For j
ahx j  1 all of the eigenvalues have modulus less than or equal to unity and hence the method is stable independent of the sign of a. The quantity j
ahx j is known as the Courant (or CFL) number. It is equal to the ratio of the distance travelled by a wave in one time step to the mesh spacing.

208

CHAPTER 11. NUMERICAL DISSIPATION

The nature of the dissipative properties of the Lax-Wendro scheme can be seen by examining the modi ed partial dierential equation, which is given by @u + a @u = ; a (
x2 ; a2h2 ) @ 3 u ; a2 h (
x2 ; a2 h2) @ 4 u + : : : @t @x 6 @x3 8 @x4 This is derived by substituting Taylor series expansions for all terms in Eq. 11.9 and converting the time derivatives to space derivatives using Eq. 11.8. The two leading error terms appear on the right side of the equation. Recall that the odd derivatives on the right side lead to unwanted dispersion and the even derivatives lead to dissipation (or ampli cation, depending on the sign). Therefore, the leading error term in the Lax-Wendro method is dispersive and proportional to 3u 2 3u a @ a
x @ 2 2 2 2 ; 6 (
x ; a h ) @x3 = ; 6 (1 ; Cn) @x3 The dissipative term is proportional to 2 4 2 2 @4u ; a8h (
x2 ; a2h2 ) @@xu4 = ; a h8
x (1 ; Cn2 ) @x 4 This term has the appropriate sign and hence the scheme is truly dissipative as long as Cn  1. A closely related method is that of MacCormack. Recall MacCormack's timemarching method, presented in Chapter 6:

u~n+1 = un + hu0n (11.10) un+1 = 21 [un + u~n+1 + hu~0n+1] If we use rst-order backward dierencing in the rst stage and rst-order forward dierencing in the second stage,1 a dissipative second-order method is obtained. For the linear convection equation, this approach leads to u~(jn+1) = u(jn) ;
ahx (u(jn) ; u(jn;)1) u(jn+1) = 12 [u(jn) + u~(jn+1) ;
ahx (~u(jn+1+1) ; u~(jn+1))] (11.11) which can be shown to be identical to the Lax-Wendro method. Hence MacCormack's method has the same dissipative and dispersive properties as the Lax-Wendro method. The two methods dier when applied to nonlinear hyperbolic systems, however. 1 Or vice-versa; for nonlinear problems, these should be applied alternately.

11.4. UPWIND SCHEMES

11.4 Upwind Schemes

209

In Section 11.1, we saw that numerical dissipation can be introduced in the spatial dierence operator using one-sided dierence schemes or, more generally, by adding a symmetric component to the spatial operator. With this approach, the direction of the one-sided operator (i.e., whether it is a forward or a backward dierence) or the sign of the symmetric component depends on the sign of the wave speed. When a hyperbolic system of equations is being solved, the wave speeds can be both positive and negative. For example, the eigenvalues of the ux Jacobian for the onedimensional Euler equations are u; u + a; u ; a. When the ow is subsonic, these are of mixed sign. In order to apply one-sided dierencing schemes to such systems, some form of splitting is required. This is avoided in the Lax-Wendro scheme. However, as a result of their superior exibility, schemes in which the numerical dissipation is introduced in the spatial operator are generally preferred over the Lax-Wendro approach. Consider again the linear convection equation: @u + a @u = 0 (11.12) @t @x where we do not make any assumptions as to the sign of a. We can rewrite Eq. 11.12 as @u + (a+ + a; ) @u = 0 ; a = a  jaj @t @x 2 + ; If a  0, then a = a  0 and a = 0. Alternatively, if a  0, then a+ = 0 and a; = a  0. Now for the a+ ( 0) term we can safely backward dierence and for the a; ( 0) term forward dierence. This is the basic concept behind upwind methods, that is, some decomposition or splitting of the uxes into terms which have positive and negative characteristic speeds so that appropriate dierencing schemes can be chosen. In the next two sections, we present two splitting techniques commonly used with upwind methods. These are by no means unique. The above approach to obtaining a stable discretization independent of the sign of a can be written in a dierent, but entirely equivalent, manner. From Eq. 11.2, we see that a stable discretization is obtained with = 1 if a  0 and with = ;1 if a  0. This is achieved by the following point operator: ;1 [a(;u + u ) + jaj(;u + 2u ; u )] ;a(xu)j = 2
(11.13) j ;1 j +1 j ;1 j j +1 x This approach is extended to systems of equations in Section 11.5. In this section, we present the basic ideas of ux-vector and ux-dierence splitting. For more subtle aspects of implementation and application of such techniques to

210

CHAPTER 11. NUMERICAL DISSIPATION

nonlinear hyperbolic systems such as the Euler equations, the reader is referred to the literature on this subject.

11.4.1 Flux-Vector Splitting Recall from Section 2.5 that a linear, constant-coecient, hyperbolic system of partial dierential equations given by

@u + @f = @u + A @u = 0 (11.14) @t @x @t @x can be decoupled into characteristic equations of the form @wi +  @wi = 0 (11.15) i @t @x where the wave speeds, i, are the eigenvalues of the Jacobian matrix, A, and the wi's are the characteristic variables. In order to apply a one-sided (or biased) spatial dierencing scheme, we need to apply a backward dierence if the wave speed, i, is positive, and a forward dierence if the wave speed is negative. To accomplish this, let us split the matrix of eigenvalues, , into two components such that  = + + ; where

+ =  +2 jj ;

; =  ;2 jj

(11.16) (11.17)

With these de nitions, + contains the positive eigenvalues and ; contains the negative eigenvalues. We can now rewrite the system in terms of characteristic variables as @w +  @w = @w + + @w + ; @w = 0 (11.18) @t @x @t @x @x The spatial terms have been split into two components according to the sign of the wave speeds. We can use backward dierencing for the + @w @x term and forward dierencing for the ; @w term. Premultiplying by X and inserting the product @x ; 1 X X in the spatial terms gives

@Xw + @X +X ;1Xw + @X ;X ;1Xw = 0 @t @x @x

(11.19)

11.4. UPWIND SCHEMES

211

With the de nitions2

A+ = X +X ;1 ;

A; = X ;X ;1

(11.20)

and recalling that u = Xw, we obtain

@u + @A+u + @A; u = 0 @t @x @x Finally the split ux vectors are de ned as f + = A+u;

(11.21)

f ; = A; u

(11.22)

and we can write

@u + @f + + @f ; = 0 (11.23) @t @x @x In the linear case, the de nition of the split uxes follows directly from the de nition of the ux, f = Au. For the Euler equations, f is also equal to Au as a result of their homogeneous property, as discussed in Appendix C. Note that f = f+ + f;

(11.24)

Thus by applying backward dierences to the f + term and forward dierences to the f ; term, we are in eect solving the characteristic equations in the desired manner. This approach is known as ux-vector splitting. When an implicit time-marching method is used, the Jacobians of the split ux vectors are required. In the nonlinear case,

@f + 6= A+ ; @f ; 6= A; @u @u Therefore, one must nd and use the new Jacobians given by +

A++ = @f @u ;

;

A;; = @f @u

(11.25)

(11.26)

For the Euler equations, A++ has eigenvalues which are all positive, and A;; has all negative eigenvalues. 2 With these de nitions A+ has all positive eigenvalues, and A; has all negative eigenvalues.

212

CHAPTER 11. NUMERICAL DISSIPATION

11.4.2 Flux-Dierence Splitting Another approach, more suited to nite-volume methods, is known as ux-dierence splitting. In a nite-volume method, the uxes must be evaluated at cell boundaries. We again begin with the diagonalized form of the linear, constant-coecient, hyperbolic system of equations @w +  @w = 0 (11.27) @t @x The ux vector associated with this form is g = w. Now, as in Chapter 5, we consider the numerical ux at the interface between nodes j and j + 1, g^j+1=2, as a function of the states to the left and right of the interface, wL and wR, respectively. The centered approximation to gj+1=2, which is nondissipative, is given by g^j+1=2 = 21 (g(wL) + g(wR)) (11.28) In order to obtain a one-sided upwind approximation, we require ( 0 (^gi)j+1=2 = i((wwi))L ifif i > (11.29) < i i R i 0 where the subscript i indicates individual components of w and g. This is achieved with (^gi)j+1=2 = 21 i [(wi)L + (wi)R ] + 21 jij [(wi)L ; (wi)R] (11.30) or g^j+1=2 = 12  (wL + wR) + 21 jj (wL ; wR ) (11.31) Now, as in Eq. 11.19, we premultiply by X to return to the original variables and insert the product X ;1X after  and jj to obtain X g^j+1=2 = 21 X X ;1X (wL + wR) + 21 X jjX ;1X (wL ; wR ) (11.32) and thus f^j+1=2 = 12 (fL + fR ) + 21 jAj (uL ; uR) (11.33) where

jAj = X jjX ;1

(11.34)

11.5. ARTIFICIAL DISSIPATION

213

and we have also used the relations f = Xg, u = Xw, and A = X X ;1. In the linear, constant-coecient case, this leads to an upwind operator which is identical to that obtained using ux-vector splitting. However, in the nonlinear case, there is some ambiguity regarding the de nition of jAj at the cell interface j + 1=2. In order to resolve this, consider a situation in which the eigenvalues of A are all of the same sign. In this case, we would like our de nition of f^j+1=2 to satisfy

(

i0s > 0 f^j+1=2 = ffL ifif all (11.35) 0 R all i s < 0 giving pure upwinding. If the eigenvalues of A are all positive, jAj = A; if they are all negative, jAj = ;A. Hence satisfaction of Eq. 11.35 is obtained by the de nition f^j+1=2 = 21 (fL + fR ) + 12 jAj+1=2j (uL ; uR) (11.36) if Aj+1=2 satis es fL ; fR = Aj+1=2 (uL ; uR) (11.37) For the Euler equations for a perfect gas, Eq. 11.37 is satis ed by the ux Jacobian evaluated at the Roe-average state given by pLuL + pR uR (11.38) uj+1=2 = p + p L R p H + p H L L Hj+1=2 = (11.39) pL + pRR R where u and H = (e + p)= are the velocity and the total enthalpy per unit mass, respectively.3

11.5 Arti cial Dissipation We have seen that numerical dissipation can be introduced by using one-sided differencing schemes together with some form of ux splitting. We have also seen that such dissipation can be introduced by adding a symmetric component to an antisymmetric (dissipation-free) operator. Thus we can generalize the concept of upwinding to include any scheme in which the symmetric portion of the operator is treated in such a manner as to be truly dissipative. 3 Note that the ux Jacobian can be written in terms of u and H only; see problem 6 at the end

of this chapter.

214

CHAPTER 11. NUMERICAL DISSIPATION

For example, let

(11.40) (xa u)j = uj+1 ; uj;1 ; (xs u)j = ;uj+1 + 2uj ; uj;1 2
x 2
x Applying x = xa + xs to the spatial derivative in Eq. 11.15 is stable if i  0 and unstable if i < 0. Similarly, applying x = xa ; xs is stable if i  0 and unstable if i > 0. The appropriate implementation is thus ix = ixa + jijxs (11.41) Extension to a hyperbolic system by applying the above approach to the characteristic variables, as in the previous two sections, gives x (Au) = xa (Au) + xs (jAju) (11.42) or xf = xa f + xs (jAju) (11.43) where jAj is de ned in Eq. 11.34. The second spatial term is known as arti cial dissipation. It is also sometimes referred to as arti cial diusion or arti cial viscosity. With appropriate choices of xa and xs , this approach can be related to the upwind approach. This is particularly evident from a comparison of Eqs. 11.36 and 11.43. It is common to use the following operator for xs (xs u)j =  (uj;2 ; 4uj;1 + 6uj ; 4uj+1 + uj+2) (11.44)
x where  is a problem-dependent coecient. This symmetric operator approximates 
x3 uxxxx and thus introduces a third-order dissipative term. With an appropriate value of , this often provides sucient damping of high frequency modes without greatly aecting the low frequency modes. For details of how this can be implemented for nonlinear hyperbolic systems, the reader should consult the literature. A more complicated treatment of the numerical dissipation is also required near shock waves and other discontinuities, but is beyond the scope of this book.

11.6 Problems 1. A second-order backward dierence approximation to a 1st derivative is given as a point operator by (xu)j = 1 (uj;2 ; 4uj;1 + 3uj ) 2
x

11.6. PROBLEMS

215

(a) Express this operator in banded matrix form (for periodic boundary conditions), then derive the symmetric and skew-symmetric matrices that have the matrix operator as their sum. (See Appendix A.3 to see how to construct the symmetric and skew-symmetric components of a matrix.) (b) Using a Taylor table, nd the derivative which is approximated by the corresponding symmetric and skew-symmetric operators and the leading error term for each. 2. Find the modi ed wavenumber for the rst-order backward dierence operator. Plot the real and imaginary parts of 
x vs. 
x for 0  
x  . Using Fourier analysis as in Section 6.6.2, nd jj for the combination of this spatial operator with 4th-order Runge-Kutta time marching at a Courant number of unity and plot vs. 
x for 0  
x  . 3. Find the modi ed wavenumber for the operator given in Eq. 11.6. Plot the real and imaginary parts of 
x vs. 
x for 0  
x  . Using Fourier analysis as in Section 6.6.2, nd jj for the combination of this spatial operator with 4th-order Runge-Kutta time marching at a Courant number of unity and plot vs. 
x for 0  
x  . 4. Consider the spatial operator obtained by combining second-order centered differences with the symmetric operator given in Eq. 11.44. Find the modi ed wavenumber for this operator with  = 0; 1=12; 1=24, and 1=48. Plot the real and imaginary parts of 
x vs. 
x for 0  
x  . Using Fourier analysis as in Section 6.6.2, nd jj for the combination of this spatial operator with 4th-order Runge-Kutta time marching at a Courant number of unity and plot vs. 
x for 0  
x  . 5. Consider the hyperbolic system derived in problem 8 of Chapter 2. Find the matrix jAj. Form the plus-minus split ux vectors as in Section 11.4.1. 6. Show that the ux Jacobian for the 1-D Euler equations can be written in terms of u and H . Show that the use of the Roe average state given in Eqs. 11.38 and 11.39 leads to satisfaction of Eq. 11.37.

216

CHAPTER 11. NUMERICAL DISSIPATION

Chapter 12 SPLIT AND FACTORED FORMS In the next two chapters, we present and analyze split and factored algorithms. This gives the reader a feel for some of the modi cations which can be made to the basic algorithms in order to obtain ecient solvers for practical multidimensional applications, and a means for analyzing such modi ed forms.

12.1 The Concept Factored forms of numerical operators are used extensively in constructing and applying numerical methods to problems in uid mechanics. They are the basis for a wide variety of methods variously known by the labels \hybrid", \time split", and \fractional step". Factored forms are especially useful for the derivation of practical algorithms that use implicit methods. When we approach numerical analysis in the light of matrix derivative operators, the concept of factoring is quite simple to present and grasp. Let us start with the following observations: 1. Matrices can be split in quite arbitrary ways. 2. Advancing to the next time level always requires some reference to a previous one. 3. Time marching methods are valid only to some order of accuracy in the step size, h. Now recall the generic ODE's produced by the semi-discrete approach

d~u = A~u ; ~f dt 217

(12.1)

CHAPTER 12. SPLIT AND FACTORED FORMS

218

and consider the above observations. From observation 1 (arbitrary splitting of A) :

d~u = [A + A ]~u ; ~f dt 1

(12.2)

2

where A = [A + A ] but A and A are not unique. For the time march let us choose the simple, rst-order, explicit Euler method. Then, from observation 2 (new data ~un in terms of old ~un): 1

2

1

1

2

+1

~un = [ I + hA + hA ]~un ; h~f + O(h ) +1

1

(12.3)

2

2

or its equivalent

~un = h[ I + hA ][ I + hA ] ; h A A i~un ; h~f + O(h ) +1

1

2

2

1

2

2

Finally, from observation 3 (allowing us to drop higher order terms ;h A A ~un): 2

1

2

~un = [ I + hA ][ I + hA ]~un ; h~f + O(h )

(12.4) Notice that Eqs. 12.3 and 12.4 have the same formal order of accuracy and, in this sense, neither one is to be preferred over the other. However, their numerical stability can be quite dierent, and techniques to carry out their numerical evaluation can have arithmetic operation counts that vary by orders of magnitude. Both of these considerations are investigated later. Here we seek only to apply to some simple cases the concept of factoring. +1

1

2

2

12.2 Factoring Physical Representations | Time Splitting Suppose we have a PDE that represents both the processes of convection and dissipation. The semi-discrete approach to its solution might be put in the form

d~u = A ~u + A ~u + (bc~ ) c d dt

(12.5)

where Ac and Ad are matrices representing the convection and dissipation terms, respectively; and their sum forms the A matrix we have considered in the previous sections. Choose again the explicit Euler time march so that

~ ) + O (h ) ~un = [ I + hAd + hAc]~un + h(bc +1

1 Second-order time-marching methods are considered later.

2

(12.6)

12.2. FACTORING PHYSICAL REPRESENTATIONS | TIME SPLITTING 219 Now consider the factored form

~un



+1



~) = [ I + hAd ] [ I + hAc]~un + h(bc ~ ) +O(h ) (12.7) ~ ) + h Ad Ac~un + (bc = [| I + hAd + h{zAc]~un + h(bc } | {z } Original Unfactored Terms Higher Order Terms 2

2

and we see that Eq. 12.7 and the original unfactored form Eq. 12.6 have identical orders of accuracy in the time approximation. Therefore, on this basis, their selection is arbitrary. In practical applications equations such as 12.7 are often applied in a predictor-corrector sequence. In this case one could write 2

u~n ~un

+1 +1

~) = [ I + hAc]~un + h(bc = [ I + hAd]~un

(12.8)

+1

Factoring can also be useful to form split combinations of implicit and explicit techniques. For example, another way to approximate Eq. 12.6 with the same order of accuracy is given by the expression 



~) = [ I ; hAd]; [ I + hAc]~un + h(bc ~ ) +O(h ) = [| I + hAd + h{zAc]~un + h(bc } Original Unfactored Terms where in this approximation we have used the fact that ~un

1

+1

2

(12.9)

[ I ; hAd ]; = I + hAd + h Ad +


1

2

2

if h
jjAdjj < 1, where jjAd jj is some norm of [Ad ]. This time a predictor-corrector interpretation leads to the sequence

u~n [ I ; hAd]~un

+1 +1

~) = [ I + hAc]~un + h(bc = u~n +1

(12.10)

The convection operator is applied explicitly, as before, but the diusion operator is now implicit, requiring a tridiagonal solver if the diusion term is central dierenced. Since numerical stiness is generally much more severe for the diusion process, this factored form would appear to be superior to that provided by Eq. 12.8. However, the important aspect of stability has yet to be discussed. 2 We do not suggest that this particular method is suitable for use. We have yet to determine its

stability, and a rst-order time-march method is usually unsatisfactory.

CHAPTER 12. SPLIT AND FACTORED FORMS

220

We should mention here that Eq. 12.9 can be derived for a dierent point of view by writing Eq. 12.6 in the form

un ; un = A u + A u + (bc ~ ) + O(h ) c n d n h +1

Then

2

+1

~) [ I ; hAd ]un = [ I + hAc]un + h(bc which is identical to Eq. 12.10. +1

12.3 Factoring Space Matrix Operators in 2{D

12.3.1 Mesh Indexing Convention

Factoring is widely used in codes designed for the numerical solution of equations governing unsteady two- and three-dimensional ows. Let us study the basic concept of factoring by inspecting its use on the linear 2-D scalar PDE that models diusion:

@u = @ u + @ u @t @x @y 2

2

2

(12.11)

2

We begin by reducing this PDE to a coupled set of ODE's by dierencing the space derivatives and inspecting the resulting matrix operator. A clear description of a matrix nite-dierence operator in 2- and 3-D requires some reference to a mesh. We choose the 3  4 point mesh shown in the Sketch 12.12. In this example Mx , the number of (interior) x points, is 4 and My , the number of (interior) y points is 3. The numbers 11 ; 12 ;


; 43 represent the location in the mesh of the dependent variable bearing that index. Thus u represents the value of u at j = 3 and k = 2. 3

32



My 13 23 33 43 k 12 22 32 42 1

11 21 31 41 1 j


Mx

(12.12)

Mesh indexing in 2-D. 3 This could also be called a 5  6 point mesh if the boundary points (labeled

in the sketch) were included, but in these notes we describe the size of a mesh by the number of interior points.

12.3. FACTORING SPACE MATRIX OPERATORS IN 2{D

221

12.3.2 Data Bases and Space Vectors

The dimensioned array in a computer code that allots the storage locations of the dependent variable(s) is referred to as a data-base. There are many ways to lay out a data-base. Of these, we consider only two: (1), consecutively along rows that are themselves consecutive from k = 1 to My , and (2), consecutively along columns that are consecutive from j = 1 to Mx. We refer to each row or column group as a space vector (they represent data along lines that are continuous in space) and label their sum with the symbol U . In particular, (1) and (2) above are referred to as xvectors and y-vectors, respectively. The symbol U by itself is not enough to identify the structure of the data-base and is used only when the structure is immaterial or understood. To be speci c about the structure, we label a data{base composed of x-vectors with U x , and one composed of y-vectors with U y . Examples of the order of indexing for these space vectors are given in Eq. 12.16 part a and b. ( )

( )

12.3.3 Data Base Permutations

The two vectors (arrays) are related by a permutation matrix P such that

U x = Pxy U y ( )

( )

U y = PyxU x

and

( )

where

(12.13)

( )

Pyx = PTxy = P;xy Now consider the structure of a matrix nite-dierence operator representing 3point central-dierencing schemes for both space derivatives in two dimensions. When the matrix is multiplying a space vector U , the usual (but ambiguous) representation is given by Ax y . In this notation the ODE form of Eq. 12.11 can be written 1

4

+

dU = A U + (bc ~) x y dt

(12.14)

+

If it is important to be speci c about the data-base structure, we use the notation Axx y or Axy y , depending on the data{base chosen for the U it multiplies. Examples are in Eq. 12.16 part a and b. Notice that the matrices are not the same although they represent the same derivative operation. Their structures are similar, however, and they are related by the same permutation matrix that relates U x to U y . Thus ( ) +

( ) +

( )

Axx y = Pxy
Axy y
Pyx ( ) +

4 Notice that

( ) +

( )

(12.15)

Ax+y and U , which are notations used in the special case of space vectors, are subsets of A and ~u, used in the previous sections.

CHAPTER 12. SPLIT AND FACTORED FORMS

222 2  6 x 6 6 6 6 6 6 6 6 6 6 o 6 6 A(xx+)y
U (x) = 666 6 6 6 6 6 6 6 6 6 6 6 4

j o  x j o x  x j o x  j o

j j j j

j  x o j x  x o j x  x o j x 

j o j o j o j

x

j o j o j o j o

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
7 7 o 777 7 7 7 7 7 7 x 75

j  x j x  x j x  j x 

11 21 31 41

;;

;;

a:Elements in 2-dimensional, central-dierence, matrix operator, Ax y , for 34 mesh shown in Sketch 12.12. Data base composed of My x{vectors stored in U x . Entries for x ! x, for y ! o, for both ! . +

2  6 o 6 6 6 6 6 6 6 x 6 6 6 6 6 6 (y ) (y ) Ax+y
U = 666 6 6 6 6 6 6 6 6 6 6 6 6 4

13 23 33 43

( )

j x j  o j x j o  j x j o

j j j

j  o j x j x j o  o j x j x j o  j x j j x j  o j x j x j o  o j x j x j o  j j j j

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 x 777 7 7 7 7 7 o5

j x j  o j x j o  j x j o 

b: Elements in 2-dimensional, central-dierence, matrix operator, Ax y , for 34 mesh shown in Sketch 12.12. Data base composed of Mx y{vectors stored in U y . Entries for x ! x, for y ! o, for both ! . +

12 22 32 42

( )

11 12 13

;; 21 22 23

;; 31 32 33

;; 41 42 43

(12.16)

12.3. FACTORING SPACE MATRIX OPERATORS IN 2{D

223

12.3.4 Space Splitting and Factoring We are now prepared to discuss splitting in two dimensions. It should be clear that the matrix Axx y can be split into two matrices such that ( ) +

Axx y = Axx + Ayx ( ) +

( )

(12.17)

( )

where Axx and Ayx are shown in Eq. 12.22. Similarily ( )

( )

Axy y = Axy + Ayy ( ) +

( )

(12.18)

( )

where the split matrices are shown in Eq. 12.23. The permutation relation also holds for the split matrices so

Ayx = Pxy Ayy Pyx ( )

and

( )

Axx = Pxy Axy Pyx ( )

( )

The splittings in Eqs. 12.17 and 12.18 can be combined with factoring in the manner described in Section 12.2. As an example ( rst-order in time), applying the implicit Euler method to Eq. 12.14 gives h i Unx = Unx + h Axx + Ayx Unx + h(bc~ ) ( ) +1

( )

( )

( )

( ) +1

or h

i I ; hAxx ; hAyx Unx = Unx + h(bc~ ) + O(h ) ( )

( ) +1

( )

( )

2

(12.19)

As in Section 12.2, we retain the same rst order accuracy with the alternative h

ih i ~ ) + O(h ) I ; hAxx I ; hAyx Unx = Unx + h(bc ( )

( ) +1

( )

( )

2

(12.20)

Write this in predictor-corrector form and permute the data base of the second row. There results h

i

~) I ; hAxx U~ x = Unx + h(bc h i y I ; hAyy Un = U~ y ( )

( )

( )

( )

( ) +1

( )

(12.21)

CHAPTER 12. SPLIT AND FACTORED FORMS

224

2 x 6 x 6 6 6 6 6 6 6 6 6 6 6 6 (x) (x) Ax
U = 666 6 6 6 6 6 6 6 6 6 6 6 4

x x x x x x x x

j j j j

j j j j

j x x j x x x j x x x j x x

j j j j

j j j j 2 o j o 6 o j o 6 6 6 o j o 6 6 o j o 6 6

6 6 6 o 6 6 A(yx)
U (x) = 666 6 6 6 6 6 6 6 6 6 6 6 4

j o o j o o j o o j o j o j o j o j o The splitting of Axx y . ( ) +

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
U (x) 7 7 7 7 7 7 7 7 7 7 7 x 75

j x x j x x x j x x j x x 3 j 7 j 7 7 7 j 7 7 j 7 7

(12.22)

7

7 7 j o 7 7 j o 7 x j o 777
U j o 777

( )

7

7 7 j o 7 7 j o 7 j o 75 j o

12.3. FACTORING SPACE MATRIX OPERATORS IN 2{D

2 x 6 6 6 6 6 6 6 6 x 6 6 6 6 6 6 A(xy)
U (y) = 666 6 6 6 6 6 6 6 6 6 6 6 6 4

j x j x j x j x j x j

j x j x j x j x j x j x j x j x j j x j x j x j x j x j x j x j x j

j j j 2 o o j 6 o o o j 6 6 o o j 6 6

6 6 6 6 6 6 6 6 6 (y ) (y ) Ay
U = 666 6 6 6 6 6 6 6 6 6 6 6 6 4

j j j

225

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
U (y) 7 7 7 7 7 x 777 7 7 7 7 7 5

j x j x j x j x j x j x 3 j j 7 j j 7 7 j j 7 7

j o o j j o o o j j o o j

j j j

j j j

j o o j j o o o j j o o j

j j j

j j j

7 7 7 7 7 7 7 7 7 7 7
U (y) 7 7 7 7 7 7 7 7 7 7 7 7 7 o5

j o o j o o j o o

The splitting of Axy y . ( ) +

(12.23)

CHAPTER 12. SPLIT AND FACTORED FORMS

226

12.4 Second-Order Factored Implicit Methods

Second-order accuracy in time can be maintained in a certain factored implicit methods. For example, apply the trapezoidal method to Eq. 12.14 where the derivative operators have been split as in Eq. 12.17 or 12.18. Let the data base be immaterial ~ ) be time invariant. There results and the (bc     1 1 1 1 ~ ) + O(h ) (12.24) I ; 2 hAx ; 2 hAy Un = I + 2 hAx + 2 hAy Un + h(bc Factor both sides giving     1 1 1 I ; 2 hAx I ; 2 hAy ; 4 h AxAy Un     1 1 1 ~ ) + O(h ) (12.25) = I + hAx I + hAy ; h AxAy Un + h(bc 2 2 4 Then notice that the combination 14 h [AxAy ](Un ; Un) is proportional to h since the leading term in the expansion of (Un ; Un) is proportional to h. Therefore, we can write       1 1 1 1 I ; 2 hAx I ; 2 hAy Un = I + 2 hAx I + 2 hAy Un + h(bc~ ) + O(h )(12.26) and both the factored and unfactored form of the trapezoidal method are second-order accurate in the time march. An alternative form of this kind of factorization is the classical ADI (alternating direction implicit) method usually written     1 1 ~ I ; 2 hAx U = I + 2 hAy Un + 12 hFn     1 1 (12.27) I ; 2 hAy Un = I + 2 hAx U~ + 12 hFn + O(h ) 3

+1

2

+1

2

2

3

3

+1

+1

3

+1

5

+1

+1

3

For idealized commuting systems the methods given by Eqs. 12.26 and 12.27 dier only in their evaluation of a time-dependent forcing term.

12.5 Importance of Factored Forms in 2 and 3 Dimensions When the time-march equations are sti and implicit methods are required to permit reasonably large time steps, the use of factored forms becomes a very valuable tool 5 A form of the Douglas or Peaceman-Rachford methods.

12.5. IMPORTANCE OF FACTORED FORMS IN 2 AND 3 DIMENSIONS 227 for realistic problems. Consider, for example, the problem of computing the time advance in the unfactored form of the trapezoidal method given by Eq. 12.24     1 1 ~) I ; 2 hAx y Un = I + 2 hAx y Un + h(bc Forming the right hand side poses no problem, but nding Un requires the solution of a sparse, but very large, set of coupled simultaneous equations having the matrix form shown in Eq. 12.16 part a and b. Furthermore, in real cases involving the Euler or Navier-Stokes equations, each symbol (o; x; ) represents a 4  4 block matrix with entries that depend on the pressure, density and velocity eld. Suppose we were to solve the equations directly. The forward sweep of a simple Gaussian elimination lls all of the 4  4 blocks between the main and outermost diagonal (e.g. between  and o in Eq. 12.16 part b.). This must be stored in computer memory to be used to nd the nal solution in the backward sweep. If Ne represents the order of the small block matrix (4 in the 2-D Euler case), the approximate memory requirement is (Ne  My )
(Ne  My )
Mx

oating point words. Here it is assumed that My < Mx. If My > Mx, My and Mx would be interchanged. A moderate mesh of 60  200 points would require over 11 million words to nd the solution. Actually current computer power is able to cope rather easily with storage requirements of this order of magnitude. With computing speeds of over one giga op, direct solvers may become useful for nding steady-state solutions of practical problems in two dimensions. However, a three-dimensional solver would require a memory of approximatly +

+1

+

+1

6

7

8

Ne
My
Mz
Mx 2

2

2

words and, for well resolved ow elds, this probably exceeds memory availability for some time to come. On the other hand, consider computing a solution using the factored implicit equation 12.25. Again computing the right hand side poses no problem. Accumulate the result of such a computation in the array (RHS ). One can then write the remaining terms in the two-step predictor-corrector form   I ; 21 hAxx U~ x = (RHS ) x   I ; 12 hAyy Uny = U~ y (12.28) ( )

( )

( )

( ) +1

( )

( )

6 For matrices as small as those shown there are many gaps in this \ ll", but for meshes of

practical size the ll is mostly dense. 7 The lower band is also computed but does not have to be saved unless the solution is to be repeated for another vector. 8 One billion oating-point operations per second.

CHAPTER 12. SPLIT AND FACTORED FORMS

228

which has the same appearance as Eq. 12.21 but is second-order time accurate. The rst step would be solved using My uncoupled block tridiagonal solvers . Inspecting the top of Eq. 12.22, we see that this is equivalent to solving My one-dimensional problems, each with Mx blocks of order Ne . The temporary solution U~ x would then be permuted to U~ y and an inspection of the bottom of Eq. 12.23 shows that the nal step consists of solving Mx one-dimensional implicit problems each with dimension My . 9

( )

( )

12.6 The Delta Form Clearly many ways can be devised to split the matrices and generate factored forms. One way that is especially useful, for ensuring a correct steady-state solution in a converged time-march, is referred to as the \delta form" and we develop it next. Consider the unfactored form of the trapezoidal method given by Eq. 12.24, and ~ ) be time invariant: let the (bc     1 1 1 1 ~ ) + O(h ) I ; 2 hAx ; 2 hAy Un = I + 2 hAx + 2 hAy Un + h(bc From both sides subtract   1 1 I ; 2 hAx ; 2 hAy Un leaving the equality unchanged. Then, using the standard de nition of the dierence operator
,
Un = Un ; Un one nds   h 1 1 ~ )i + O(h ) I ; 2 hAx ; 2 hAy
Un = h Ax y Un + (bc (12.29) Notice that the right side of this equation is the product of h and a term that is identical to the right side of Eq. 12.14, our original ODE. Thus, if Eq. 12.29 converges, it is guaranteed to converge to the correct steady-state solution of the ODE. Now we can factor Eq. 12.29 and maintain O(h ) accuracy. We arrive at the expression    h i 1 1 (12.30) I ; 2 hAx I ; 2 hAy
Un = h Ax y Un + (bc~ ) + O(h ) This is the delta form of a factored, 2nd-order, 2-D equation. 3

+1

+1

3

+

2

+

3

9 A block tridiagonal solver is similar to a scalar solver except that small block matrix operations

replace the scalar ones, and matrix multiplications do not commute.

12.7. PROBLEMS

229

The point at which the factoring is made may not aect the order of time-accuracy, but it can have a profound eect on the stability and convergence properties of a method. For example, the unfactored form of a rst-order method derived from the implicit Euler time march is given by Eq. 12.19, and if it is immediately factored, the factored form is presented in Eq. 12.20. On the other hand, the delta form of the unfactored Eq. 12.19 is h

i

~) [I ; hAx ; hAy ]
Un = h Ax y Un + (bc +

and its factored form becomes

10

h

~) [I ; hAx][I ; hAy ]
Un = h Ax y Un + (bc

i

+

(12.31)

In spite of the similarities in derivation, we will see in the next chapter that the convergence properties of Eq. 12.20 and Eq. 12.31 are vastly dierent.

12.7 Problems 1. Consider the 1-D heat equation:

@u =  @ u @t @x 2

0x9

2

Let u(0; t) = 0 and u(9; t) = 0, so that we can simplify the boundary conditions. Assume that second order central dierencing is used, i.e., (xxu)j = 1 (uj; ; 2uj + uj )
x The uniform grid has
x = 1 and 8 interior points. 2

1

+1

(a) Space vector de nition i. What is the space vector for the natural ordering (monotonically increasing in index), u ? Only include the interior points. ii. If we reorder the points with the odd points rst and then the even points, write the space vector, u ? iii. Write down the permutation matrices,(P ,P ). (1)

(2)

12

21

10 Notice that the only dierence between the O(h2 ) method given by Eq. 12.30 and the O(h) method given by Eq. 12.31 is the appearance of the factor 1 on the left side of the O(h2 ) method.

2

CHAPTER 12. SPLIT AND FACTORED FORMS

230

iv. The generic ODE representing the discrete form of the heat equation is

du = A u + f dt Write down the matrix A . (Note f = 0, due to the boundary conditions) Next nd the matrix A such that du = A u dt Note that A can be written as 2 T3 U D 7 A = 64 5 U D De ne D and U . (1)

(1)

1

1

2

(2)

2

(2)

2

2

v. Applying implicit Euler time marching, write the delta form of the implicit algorithm. Comment on the form of the resulting implicit matrix operator. (b) System de nition In problem 1a, we de ned u ; u ; A ; A ; P , and P which partition the odd points from the even points. We can put such a partitioning to use. First de ne extraction operators 21 0 0 0 0 0 0 03 6 0 0 0 0 77 0 1 0 0 6 6 3 0 0 0 0 777 2 I 0 0 1 0 6 6 0 7 6 0 0 0 0 7 = 64 7 I o = 66 00 00 00 01 5 7 0 0 0 0 7 6 0 6 6 0 0 0 0 777 0 60 0 0 0 40 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 03 6 0 0 0 0 0 0 0 0 77 6 6 3 0 0 0 0 0 0 0 0 777 2 0 6 6 0 7 6 0 0 0 0 7 = 64 7 I e = 66 00 00 00 00 5 7 1 0 0 0 7 6 0 I 7 6 6 0 1 0 0 77 60 0 0 0 40 0 0 0 0 0 1 05 0 0 0 0 0 0 0 1 (1)

( )

( )

(2)

1

2

12

21

4

4

4

4

4

4

4

4

12.7. PROBLEMS

231

which extract the odd even points from u as follows: u o = I o u and ue =Ieu . i. Beginning with the ODE written in terms of u , de ne a splitting A = Ao + Ae, such that Ao operates only on the odd terms, and Ae operates only on the even terms. Write out the matrices Ao and Ae. Also, write them in terms of D and U de ned above. ii. Apply implicit Euler time marching to the split ODE. Write down the delta form of the algorithm and the factored delta form. Comment on the order of the error terms. iii. Examine the implicit operators for the factored delta form. Comment on their form. You should be able to argue that these are now trangular matrices (a lower and an upper). Comment on the solution process this gives us relative to the direct inversion of the original system. (2)

( )

( )

( )

(2)

(2)

2

( )

(2)

232

CHAPTER 12. SPLIT AND FACTORED FORMS

Chapter 13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS In Section 4.4 we introduced the concept of the representative equation, and used it in Chapter 7 to study the stability, accuracy, and convergence properties of timemarching schemes. The question is: Can we nd a similar equation that will allow us to evaluate the stability and convergence properties of split and factored schemes? The answer is yes | for certain forms of linear model equations. The analysis in this chapter is useful for estimating the stability and steady-state properties of a wide variety of time-marching schemes that are variously referred to as time-split, fractional-step, hybrid, and (approximately) factored. When these methods are applied to practical problems, the results found from this analysis are neither necessary nor sucient to guarantee stability. However, if the results indicate that a method has an instability, the method is probably not suitable for practical use.

13.1 The Representative Equation for Circulant Operators Consider linear PDE's with coecients that are xed in both space and time and with boundary conditions that are periodic. We have seen that under these conditions a semi-discrete approach can lead to circulant matrix dierence operators, and we discussed circulant eigensystems1 in Section 4.3. In this and the following section we assume circulant systems and our analysis depends critically on the fact that all circulant matrices commute and have a common set of eigenvectors. 1

See also the discussion on Fourier stability analysis in Section 7.7.

233

234 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS Suppose, as a result of space dierencing the PDE, we arrive at a set of ODE's that can be written d~u = A ~u + A ~u ; ~f (t) (13.1) ap bp dt where the subscript p denotes a circulant matrix. Since both matrices have the same set of eigenvectors, we can use the arguments made in Section 4.2.3 to uncouple the set and form the M set of independent equations

w10 = (a + b)1 w1 ; g1 (t) ... wm0 = (a + b)m wm ; gm(t) ... wM0 = (a + b)M wM ; gM (t)

(13.2)

The analytic solution of the m'th line is

wm(t) = cm e(a+b)mt + P:S: Note that each a pairs with one, and only one2 , b since they must share a common eigenvector. This suggests (see Section 4.4: The representative equation for split, circulant systems is du = [ +  +  +


]u + aet (13.3) a b c dt where a + b + c +


are the sum of the eigenvalues in Aa ; Ab ; Ac ;


that share the same eigenvector.

13.2 Example Analysis of Circulant Systems 13.2.1 Stability Comparisons of Time-Split Methods Consider as an example the linear convection-diusion equation: @u + a @u =  @ 2 u @t @x @x2 2

(13.4)

This is to be contrasted to the developments found later in the analysis of 2-D equations.

13.2. EXAMPLE ANALYSIS OF CIRCULANT SYSTEMS

235

If the space dierencing takes the form d~u = ; a B (;1; 0; 1)~u +  B (1; ;2; 1)~u (13.5) dt 2
x p
x2 p the convection matrix operator and the diusion matrix operator, can be represented by the eigenvalues c and d, respectively, where (see Section 4.3.2): (c)m =
iax sin m (13.6) (d)m = ; 42 sin2 2m
x In these equations m = 2m=M , m = 0 ; 1 ;


; M ; 1 , so that 0  m  2. Using these values and the representative equation 13.4, we can analyze the stability of the two forms of simple time-splitting discussed in Section 12.2. In this section we refer to these as 1. the explicit-implicit Euler method, Eq. 12.10. 2. the explicit-explicit Euler method, Eq. 12.8.

1. The Explicit-Implicit Method

When applied to Eq. 13.4, the characteristic polynomial of this method is P (E ) = (1 ; hd )E ; (1 + hc) This leads to the principal  root 1 + i
ahx sin m = 1 + 4 h2 sin2 m 2
x where we have made use of Eq. 13.6 to quantify the eigenvalues. Now introduce the dimensionless numbers Cn =
ahx ; Courant number R
= a
 x ; mesh Reynolds number and we can write for the absolute value of  q 1 + Cn2 sin2 m jj = ; 0  m  2 (13.7)  C m n 2 1 + 4 R sin 2


236 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 1.2

1.2 C = 2/ R ∆ n

C = R /2 n ∆

0.8 C

C

n

0.4

0

C = 2/ R n ∆

0.8 n

0.4

0

4 R∆ Explicit-Implicit

8

0

0

4 R∆ Explicit-Explicit

8

Figure 13.1: S_ tability regions for two simple time-split methods. A simple numerical parametric study of Eq. 13.7 shows that the critical range of m for any combination of Cn and R
occurs when m is near 0 (or 2). From this we nd that the condition on Cn and R
that make jj  1 is 2 h i  C  n 2 2 2 1 + Cn sin  = 1 + 4 R sin 2
As  ! 0 this gives the stability region Cn < R2
which is bounded by a hyperbola and shown in Fig. 13.1.

2. The Explicit-Explicit Method An analysis similar to the one given above shows that this method produces " # q C  n m 2 2 jj = 1 + Cn2 sin m 1 ; 4 R sin 2 ; 0  m  2
Again a simple numerical parametric study shows that this has two critical ranges of m , one near 0, which yields the same result as in the previous example, and the

13.2. EXAMPLE ANALYSIS OF CIRCULANT SYSTEMS

237

other near 180o, which produces the constraint that Cn < 21 R
for R
 2 The resulting stability boundary is also shown in Fig. 13.1. The totaly explicit, factored method has a much smaller region of stability when R
is small, as we should have expected.

13.2.2 Analysis of a Second-Order Time-Split Method

Next let us analyze a more practical method that has been used in serious computational analysis of turbulent ows. This method applies to a ow in which there is a combination of diusion and periodic convection. The convection term is treated explicitly using the second-order Adams-Bashforth method. The diusion term is integrated implicitly using the trapezoidal method. Our model equation is again the linear convection-diusion equation 13.4 which we split in the fashion of Eq. 13.5. In order to evaluate the accuracy, as well as the stability, we include the forcing function in the representative equation and study the eect of our hybrid, time-marching method on the equation u0 = cu + du + aet First let us nd expressions for the two polynomials, P (E ) and Q(E ). The characteristic polynomial follows from the application of the method to the homogeneous equation, thus un+1 = un + 21 hc(3un ; un;1) + 21 hd (un+1 + un) This produces P (E ) = (1 ; 12 hd)E 2 ; (1 + 23 hc + 12 hd)E + 12 hc The form of the particular polynomial depends upon whether the forcing function is carried by the AB2 method or by the trapezoidal method. In the former case it is Q(E ) = 21 h(3E ; 1) (13.8) and in the latter (13.9) Q(E ) = 12 h(E 2 + E )

238 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS

Accuracy

From the characteristic polynomial we see that there are two {roots and they are given by the equation s

2   1 + 32 hc + 21 hd ; 2hc 1 ; 21 hd   (13.10) 2 1 ; 12 hd The principal -root follows from the plus sign and one can show   1 = 1 + (c + d )h + 12 (c + d )2h2 + 41 3d + c2d ; 2c d ; 3c h3 From this equation it is clear that 61 3 = 61 (c + d)3 does not match the coecient of h3 in 1 , so er = O(h3) Using P (eh) and Q(eh ) to evaluate er in Section 6.6.3, one can show er = O(h3) using either Eq. 13.8 or Eq. 13.9. These results show that, for the model equation, the hybrid method retains the second-order accuracy of its individual components.

1 + 23 hc + 12 hd  =

Stability

The stability of the method can be found from Eq. 13.10 by a parametric study of cn and R
de ned in Eq. 13.7. This was carried out in a manner similar to that used to nd the stability boundary of the rst-order explicit-implicit method in Section 13.2.1. The results are plotted in Fig. 13.2. For values of R
 2 this second-order method has a much greater region of stability than the rst-order explicit-implicit method given by Eq. 12.10 and shown in Fig. 13.1.

13.3 The Representative Equation for Space-Split Operators Consider the 2-D model3 equations @u = @ 2 u + @ 2 u @t @x2 @y2 3

The extension of the following to 3-D is simple and straightforward.

(13.11)

13.3. THE REPRESENTATIVE EQUATION FOR SPACE-SPLIT OPERATORS239 1.2

0.8

Cn Stable

0.4

0

8

4

R

∆

_ Figure 13.2: Stability regions for the second-order time-split method. and

@u + a @u + a @u = 0 (13.12) @t x @x y @y Reduce either of these, by means of spatial dierencing approximations, to the coupled set of ODE's: dU = [A + A ]U + (bc ~) (13.13) x y dt for the space vector U . The form of the Ax and Ay matrices for three-point central dierencing schemes are shown in Eqs. 12.22 and 12.23 for the 3  4 mesh shown in Sketch 12.12. Let us inspect the structure of these matrices closely to see how we can diagonalize [Ax + Ay ] in terms of the individual eigenvalues of the two matrices considered separately. First we write these matrices in the form 2 3 2 ~ 3 ~b1
I B b
I 0 A(xx) = 64 B 75 A(yx) = 64 ~b;1
I ~b0
I ~b1
I 75 ~b;1
I ~b0
I B where B is a banded matrix of the form B (b;1 ; b0; b1 ). Now nd the block eigenvector

240 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS matrix that diagonalizes B and use it to diagonalize A(xx). Thus 2 ;1 32 32 3 2 3 X B X  6 76 7=6 X ;1 B 75 64 X  75 4 54 5 4 X ;1 B X  where 2 3 1 6 7 2 7  = 664 7 5 3 4 ( x ) Notice that the matrix Ay is transparent to this transformation. That is, if we set X  diag(X ) 2 ~ 3 2 ~ 3 b0
I ~b1
I b0
I ~b1
I X ;1 64 ~b;1
I ~b0
I ~b1
I 75 X = 64 ~b;1
I ~b0
I ~b1
I 75 ~b;1
I ~b0
I ~b;1
I ~b0
I One now permutes the transformed system to the y-vector data-base using the permutation matrix de ned by Eq. 12.13. There results h i Pyx
X ;1 A(xx) + A(yx) X
Pxy = 3 2 3 2 ~ B 1
I 6 7 6 7 ~ 6 7 
I B 6 7 2 6 7 6 7+6 (13.14) 4 5 4 3
I B~ 75 4
I B~ where B~ is the banded tridiagonal matrix B (~b;1 ; ~b0 ; ~b1), see the bottom of Eq. 12.23. Next nd the eigenvectors X~ that diagonalize the B~ blocks. Let B~  diag(B~ ) and X~  diag(X~ ) and form the second transformation 2~ 3 2~ 3   1 6 7 ~ 7 X~ ;1B~ X~ = 664  ~ 775 ; ~ = 64 ~2 5 ~ 3 ~ This time, by the same argument as before, the rst matrix on the right side of Eq. 13.14 is transparent to the transformation, so the nal result is the complete diagonalization of the matrix Ax+y : h ih ih i X~ ;1
Pyx
X ;1 A(xx+)y X
Pxy
X~ = 2 3 ~  I +  1 6 7 6 7 2 I + ~ 6 7 (13.15) 6 7 3 I + ~ 4 5 4I + ~

13.3. THE REPRESENTATIVE EQUATION FOR SPACE-SPLIT OPERATORS241 It is important to notice that:  The diagonal matrix on the right side of Eq. 13.15 contains every possible combination of the individual eigenvalues of B and B~ . Now we are ready to present the representative equation for two dimensional systems. First reduce the PDE to ODE by some choice4 of space dierencing. This results in a spatially split A matrix formed from the subsets

A(xx) = diag(B ) ; A(yy) = diag(B~ )

(13.16)

where B and B~ are any two matrices that have linearly independent eigenvectors (this puts some constraints on the choice of dierencing schemes). Although Ax and Ay do commute, this fact, by itself, does not ensure the property of \all possible combinations". To obtain the latter property the structure of the matrices is important. The block matrices B and B~ can be either circulant or noncirculant; in both cases we are led to the nal result: The 2{D representative equation for model linear systems is du = [ +  ]u + aet x y dt where x and y are any combination of eigenvalues from Ax and Ay , a and  are (possibly complex) constants, and where Ax and Ay satisfy the conditions in 13.16. Often we are interested in nding the value of, and the convergence rate to, the steady-state solution of the representative equation. In that case we set  = 0 and use the simpler form du = [ +  ]u + a (13.17) x y dt which has the exact solution u(t) = ce(x+y )t ;  +a  (13.18) x y We have used 3-point central dierencing in our example, but this choice was for convenience only, and its use is not necessary to arrive at Eq. 13.15. 4

242 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS

13.4 Example Analysis of 2-D Model Equations

In the following we analyze four dierent methods for nding a xed, steady-state solution to the 2-D representative equation 13.16. In each case we examine 1. The stability. 2. The accuracy of the xed, steady-state solution. 3. The convergence rate to reach the steady-state.

13.4.1 The Unfactored Implicit Euler Method Consider rst this unfactored, rst-order scheme which can then be used as a reference case for comparison with the various factored ones. The form of the method is given by Eq. 12.19, and when it is applied to the representative equation, we nd (1 ; h x ; h y )un+1 = un + ha from which

P (E ) = (1 ; h x ; h y )E ; 1 Q(E ) = h giving the solution

"

(13.19)

#

n 1 un = c 1 ; h  ; h  ;  +a  x y x y Like its counterpart in the 1-D case, this method:

1. Is unconditionally stable. 2. Produces the exact (see Eq. 13.18) steady-state solution (of the ODE) for any h. 3. Converges very rapidly to the steady-state when h is large. Unfortunately, however, use of this method for 2-D problems is generally impractical for reasons discussed in Section 12.5.

13.4. EXAMPLE ANALYSIS OF 2-D MODEL EQUATIONS

243

13.4.2 The Factored Nondelta Form of the Implicit Euler Method

Now apply the factored Euler method given by Eq. 12.20 to the 2-D representative equation. There results (1 ; h x)(1 ; h y )un+1 = un + ha from which P (E ) = (1 ; h x)(1 ; h y )E ; 1 Q(E ) = h (13.20) giving the solution " #n 1 un = c (1 ; h  )(1 ; h  ) ;  +  a; h  x y x y x y We see that this method: 1. Is unconditionally stable. 2. Produces a steady state solution that depends on the choice of h. 3. Converges rapidly to a steady-state for large h, but the converged solution is completely wrong. The method requires far less storage then the unfactored form. However, it is not very useful since its transient solution is only rst-order accurate and, if one tries to take advantage of its rapid convergence rate, the converged value is meaningless.

13.4.3 The Factored Delta Form of the Implicit Euler Method

Next apply Eq. 12.31 to the 2-D representative equation. One nds (1 ; h x)(1 ; h y )(un+1 ; un) = h(xun + y un + a) which reduces to   (1 ; h x)(1 ; h y )un+1 = 1 + h2xy un + ha and this has the solution " #n 2 xy 1 + h un = c (1 ; h  )(1 ; h  ) ;  +a  x y x y This method:

244 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 1. Is unconditionally stable. 2. Produces the exact steady-state solution for any choice of h. 3. Converges very slowly to the steady{state solution for large values of h, since jj ! 1 as h ! 1. Like the factored nondelta form, this method demands far less storage than the unfactored form, as discussed in Section 12.5. The correct steady solution is obtained, but convergence is not nearly as rapid as that of the unfactored form.

13.4.4 The Factored Delta Form of the Trapezoidal Method

Finally consider the delta form of a second-order time-accurate method. Apply Eq. 12.30 to the representative equation and one nds    1 1 1 ; 2 hx 1 ; 2 hy (un+1 ; un) = h(xun + y un + a) which reduces to       1 1 1 1 1 ; hx 1 ; hy un+1 = 1 + hx 1 + hy un + ha 2 2 2 2 and this has the solution 2 1 h 1 + 1 h  3n 1 + x y 7 6 un = c64  21  21  75 ;  +a  x y 1 ; 2 hx 1 ; 2 hy This method: 1. Is unconditionally stable. 2. Produces the exact steady{state solution for any choice of h. 3. Converges very slowly to the steady{state solution for large values of h, since jj ! 1 as h ! 1. All of these properties are identical to those found for the factored delta form of the implicit Euler method. Since it is second order in time, it can be used when time accuracy is desired, and the factored delta form of the implicit Euler method can be used when a converged steady-state is all that is required.5 A brief inspection of eqs. 12.26 and 12.27 should be enough to convince the reader that the 's produced by those methods are identical to the  produced by this method. In practical codes, the value of h on the left side of the implicit equation is literally switched from h to 12 h. 5

13.5. EXAMPLE ANALYSIS OF THE 3-D MODEL EQUATION

245

13.5 Example Analysis of the 3-D Model Equation

The arguments in Section 13.3 generalize to three dimensions and, under the conditions given in 13.16 with an A(zz) included, the model 3-D cases6 have the following representative equation (with  = 0): du = [ +  +  ]u + a (13.21) x y z dt Let us analyze a 2nd-order accurate, factored, delta form using this equation. First apply the trapezoidal method: un+1 = un + 12 h[(x + y + z )un+1 + (x + y + z )un + 2a] Rearrange terms:     1 ; 21 h(x + y + z ) un+1 = 1 + 21 h(x + y + z ) un + ha Put this in delta form:   1 ; 1 h(x + y + z )
un = h[(x + y + z )un + a] 2 Now factor the left side:     1 1 1 h 1 ; h 1 ; h 1; 2 x 2 y 2 z
un = h[(x + y + z )un + a] (13.22) This preserves second order accuracy since the error terms 1 h2(  +   +   )
u and 1 h3   4 x y x z y z n 8 x y z are both O(h3). One can derive the characteristic polynomial for Eq. 13.22, nd the  root, and write the solution either in the form 2 3n 1 1 1 2 3 6 1 + h(x + y + z ) + 4 h (x y + x z + y z ) ; 8 h x y z 7 7 un = c64 21 5 1 1 2 3 1 ; 2 h(x + y + z ) + 4 h (xy + xz + y z ) ; 8 h xy z

;  + a +  x y z 6

Eqs. 13.11 and 13.12, each with an additional term.

(13.23)

246 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS or in the form

   3n 2 1 1 1 1 3 6 1 + 2 hx 1 + 2 hy 1 + 2 hz ; 4 h x y z 7 a (13.24) 7 ;     un = c64 5 1 1 1 x + y + z 1 ; hx 1 ; hy 1 ; hz

2 2 2 It is interesting to notice that a Taylor series expansion of Eq. 13.24 results in  = 1 + h(x + y + z ) + 21 h2(x + y + z )2 (13.25) h   i + 14 h3 3z + (2y + 2x) + 22y + 3xy + 22y + 3y + 2x2y + 22xy + 3x +


which veri es the second order accuracy of the factored form. Furthermore, clearly, if the method converges, it converges to the proper steady-state.7 With regards to stability, it follows from Eq. 13.23 that, if all the 's are real and negative, the method is stable for all h. This makes the method unconditionally stable for the 3-D diusion model when it is centrally dierenced in space. Now consider what happens when we apply this method to the biconvection model, the 3-D form of Eq. 13.12 with periodic boundary conditions. In this case, central dierencing causes all of the 's to be imaginary with spectrums that include both positive and negative values. Remember that in our analysis we must consider every possible combination of these eigenvalues. First write the  root in Eq. 13.23 in the form + i ; + i  = 11 ; i ; + i

where , and are real numbers that can have any sign. Now we can always nd one combination of the 's for which , and are both positive. In that case since the absolute value of the product is the product of the absolute values 2 + ( + )2 (1 ; ) 2 jj = (1 ; )2 + ( ; )2

>1

and the method is unconditionally unstable for the model convection problem. From the above analysis one would come to the conclusion that the method represented by Eq. 13.22 should not be used for the 3-D Euler equations. In practical cases, however, some form of dissipation is almost always added to methods that are used to solve the Euler equations and our experience to date is that, in the presence of this dissipation, the instability disclosed above is too weak to cause trouble. 7

However, we already knew this because we chose the delta form.

13.6. PROBLEMS

13.6 Problems

247

1. Starting with the generic ODE, du = Au + f dt we can split A as follows: A = A1 + A2 + A3 + A4. Applying implicit Euler time marching gives un+1 ; un = A u + A u + A u + A u + f 1 n+1 2 n+1 3 n+1 4 n+1 h (a) Write the factored delta form. What is the error term? (b) Instead of making all of the split terms implicit, leave two explicit: un+1 ; un = A u + A u + A u + A u + f 1 n+1 2 n 3 n+1 4 n h Write the resulting factored delta form and de ne the error terms. (c) The scalar representative equation is du = ( +  +  +  )u + a 1 2 3 4 dt For the fully implicit scheme of problem 1a, nd the exact solution to the resulting scalar dierence equation and comment on the stability, convergence, and accuracy of the converged steady-state solution. (d) Repeat 1c for the explicit-implicit scheme of problem 1b.

248 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS

Appendix A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA A basic understanding of the fundamentals of linear algebra is crucial to our development of numerical methods and it is assumed that the reader is at least familar with this subject area. Given below is some notation and some of the important relations between matrices and vectors.

A.1 Notation 1. In the present context a vector is a vertical column or string. Thus 2 v1 3 6 7 ~v = 66 v..2 77 4 . 5 vm T and its transpose ~v is the horizontal row ~vT = [v1 ; v2; v3; : : : ; vm] ; ~v = [v1 ; v2; v3 ; : : : ; vm ]T 2. A general m  m matrix A can be written 2 a11 a12 6a a A = (aij ) = 664 21 22 am1 am2 249




a1m 3


a2m 777 ...




amm

5

250APPENDIX A. USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA 3. An alternative notation for A is h i A = ~a1; ~a2; : : : ; ~am and its transpose AT is 2 ~T 3 66 ~a1T 77 AT = 666 a.2 777 4 ..T 5 ~am

4. The inverse of a matrix (if it exists) is written A;1 and has the property that A;1A = AA;1 = I , where I is the identity matrix.

A.2 De nitions 1. A is symmetric if AT = A. 2. A is skew-symmetric or antisymmetric if AT = ;A. 3. A is diagonally dominant if aii  Pj6=i jaij j ; i = 1; 2; : : : ; m and aii > Pj6=i jaij j for at least one i. 4. A is orthogonal if aij are real and AT A = AAT = I 5. A is the complex conjugate of A. 6. P is a permutation matrix if P ~v is a simple reordering of ~v. 7. The trace of a matrix is Pi aii. 8. A is normal if AT A = AAT . 9. det[A] is the determinant of A. 10. AH is the conjugate transpose of A, (Hermitian). 11. If a b  A= c d then det[A] = ad ; bc and   A;1 = det[1 A] ;dc ;ab

A.3. ALGEBRA

251

A.3 Algebra We consider only square matrices of the same dimension. 1. A and B are equal if aij = bij for all i; j = 1; 2; : : : ; m. 2. A + (B + C ) = (C + A) + B , etc. 3. sA = (saij ) where s is a scalar. 4. In general AB 6= BA. 5. Transpose equalities: (A + B )T = AT + B T (AT )T = A (AB )T = B T AT 6. Inverse equalities (if the inverse exists): (A;1);1 = A (AB );1 = B ;1 A;1 (AT );1 = (A;1 )T

7. Any matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix. Thus:     A = 21 A + AT + 12 A ; AT

A.4 Eigensystems 1. The eigenvalue problem for a matrix A is de ned as A~x = ~x or [A ; I ]~x = 0 and the generalized eigenvalue problem, including the matrix B , as A~x = B~x or [A ; B ]~x = 0 2. If a square matrix with real elements is symmetric, its eigenvalues are all real. If it is asymmetric, they are all imaginary.

252APPENDIX A. USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA 3. Gershgorin's theorem: The eigenvalues of a matrix lie in the complex plane in the union of circles having centers located by the diagonals with radii equal to the sum of the absolute values of the corresponding o-diagonal row elements. 4. In general, an m  m matrix A has n~x linearly independent eigenvectors with n~x  m and n distinct eigenvalues (i) with n  n~x  m. 5. A set of eigenvectors is said to be linearly independent if a
~xm + b
~xn 6= ~xk ; m 6= n 6= k for any complex a and b and for all combinations of vectors in the set. 6. If A posseses m linearly independent eigenvectors then A is diagonalizable, i.e.,

X ;1AX =  where X is a matrix whose columns are the eigenvectors, h i X = ~x1 ; ~x2; : : : ; ~xm and  is the diagonal matrix

2 0


66 01  . . .  = 66 .. . .2 . . 4. . .

03 ... 77 7 0 75 0


0 m If A can be diagonalized, its eigenvectors completely span the space, and A is said to have a complete eigensystem.

7. If A has m distinct eigenvalues, then A is always diagonalizable, and with each distinct eigenvalue there is one associated eigenvector, and this eigenvector cannot be formed from a linear combination of any of the other eigenvectors. 8. In general, the eigenvalues of a matrix may not be distinct, in which case the possibility exists that it cannot be diagonalized. If the eigenvalues of a matrix are not distinct, but all of the eigenvectors are linearly independent, the matrix is said to be derogatory, but it can still be diagonalized. 9. If a matrix does not have a complete set of linearly independent eigenvectors, it cannot be diagonalized. The eigenvectors of such a matrix cannot span the space and the matrix is said to have a defective eigensystem.

A.4. EIGENSYSTEMS

253

10. Defective matrices cannot be diagonalized but they can still be put into a compact form by a similarity transform, S , such that

2J 0


0 3 66 01 J . . . ... 77 7 J = S ;1AS = 66 .. . .2 . . 4 . . . 0 75 0


0 Jk

where there are k linearly independent eigenvectors and Ji is either a Jordan subblock or i. 11. A Jordan submatrix has the form

2 1 0 66 0i  1 i 66 Ji = 66 0 0 i 64 ... ... 0


0




0 3 ... ... ... 0

... 77 7 0 777 1 75 i

12. Use of the transform S is known as putting A into its Jordan Canonical form. A repeated root in a Jordan block is referred to as a defective eigenvalue. For each Jordan submatrix with an eigenvalue i of multiplicity r, there exists one eigenvector. The other r ; 1 vectors associated with this eigenvalue are referred to as principal vectors. The complete set of principal vectors and eigenvectors are all linearly independent. 13. Note that if P is the permutation matrix

2 3 0 0 1 P = 64 0 1 0 75 1 0 0

then

;

P T = P ;1 = P

2 3 2 3  1 0  0 0 P ;1 64 0  1 75 P = 64 1  0 75 0 0  0 1 

14. Some of the Jordan subblocks may have the same eigenvalue. For example, the

254APPENDIX A. USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA matrix

3 2 2 1 1 66 64 1 1 75 66 1 66 1   66  1 1 66 1  66 2 1  64 2

is both defective and derogatory, having:

     

3

3 77 77 77 77 77 77 75

9 eigenvalues 3 distinct eigenvalues 3 Jordan blocks 5 linearly independent eigenvectors 3 principal vectors with 1 1 principal vector with 2

A.5 Vector and Matrix Norms 1. The spectral radius of a matrix A is symbolized by (A) such that

(A) = jm jmax where m are the eigenvalues of the matrix A. 2. A p-norm of the vector ~v is de ned as

0M 11=p X jjvjjp = @ jvj jpA j =1

3. A p-norm of a matrix A is de ned as

jjAvjjp jjAjjp = max x6=0 jjv jj p

A.5. VECTOR AND MATRIX NORMS

255

4. Let A and B be square matrices of the same order. All matrix norms must have the properties

jjAjj jjc
Ajj jjA + B jj jjA
B jj

 0; jjAjj = 0 implies A = 0 = jcj
jjAjj  jjAjj + jjB jj  jjAjj
jjB jj

5. Special p-norms are jjAjj1 = maxj=1;


;M PMi=1 jaij j maximum column sum

q T jjAjj2 =  (A
A ) jjAjj1 = maxi=1;2;


;M PMj=1 jaij j maximum row sum where jjAjjp is referred to as the Lp norm of A. 6. In general (A) does not satisfy the conditions in 4, so in general (A) is not a true norm. 7. When A is normal, (A) is a true norm, in fact, in this case it is the L2 norm. 8. The spectral radius of A, (A), is the lower bound of all the norms of A.

256APPENDIX A. USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA

Appendix B SOME PROPERTIES OF TRIDIAGONAL MATRICES B.1 Standard Eigensystem for Simple Tridiagonals In this work tridiagonal banded matrices are prevalent. It is useful to list some of their properties. Many of these can be derived by solving the simple linear dierence equations that arise in deriving recursion relations. Let us consider a simple tridiagonal matrix, i.e., a tridiagonal with constant scalar elements a,b, and c, see Section 3.4. If we examine the conditions under which the determinant of this matrix is zero, we nd (by a recursion exercise) det[B (M : a; b; c)] = 0 if   p m b + 2 ac cos M + 1 = 0 ; m = 1; 2;


; M

From this it follows at once that the eigenvalues of B (a; b; c) are   p m (B.1) m = b + 2 ac cos M + 1 ; m = 1; 2;


; M The right-hand eigenvector of B (a; b; c) that is associated with the eigenvalue m satis es the equation B (a; b; c)~xm = m~xm (B.2) and is given by  j ; 1    ~xm = (xj )m = a 2 sin j m (B.3) c M + 1 ; m = 1; 2;


; M 257

258

APPENDIX B. SOME PROPERTIES OF TRIDIAGONAL MATRICES

These vectors are the columns of the right-hand eigenvector matrix, the elements of which are  j ; 1   j = 1; 2;


; M X = (xjm) = ac 2 sin Mjm ; (B.4) m = 1; 2;


; M +1 Notice that if a = ;1 and c = 1,  j ; 1 a 2 = ei j; 2 (B.5) c (

1)

The left-hand eigenvector matrix of B (a; b; c) can be written  m ; 1   c 2 mj = 1; 2;


; M 2 ; X = M +1 a sin M + 1 ; m j = 1; 2;


; M In this case notice that if a = ;1 and c = 1  m ; 1 c 2 = e;i m; 2 a 1

(

1)

(B.6)

B.2 Generalized Eigensystem for Simple Tridiagonals This system is de ned as follows 2 32 32 x 3 2e f x 3 b c 6 7 6 7 6 x 777 666 d e f x 777 a b c 6 7 6 7 6 6 7 6 7 6 6 7 6 7 6 x 77 =  66 d e x 77 a b 6 7 6 7 6 . . . 6 7 6 7 6 7 6 . . . c 5 4 .. 5 4 . f 5 4 ... 75 4 a b xM d e xM In this case one can show after some algebra that det[B (a ; d; b ; e; c ; f ] = 0 (B.7) if   q m b ; m e + 2 (a ; md)(c ; m f ) cos M + 1 = 0 ; m = 1; 2;


; M (B.8) If we de ne m = Mm ; p = cos m +1 m 1

1

2

2

3

3

B.3. THE INVERSE OF A SIMPLE TRIDIAGONAL

259

q

m = eb ; 2(cd + af )pm + 2pm (ec ; fb)(ea ; bd) + [(cd ; af )pm ] e ; 4fdpm The right-hand eigenvectors are #j ; 1 " 2 a ;  d m = 1; 2;


; M m ~xm = sin [ j m] ; j = 1; 2;


; M c; f 2

2

2

2

m

These relations are useful in studying relaxation methods.

B.3 The Inverse of a Simple Tridiagonal The inverse of B (a; b; c) can also be written in analytic form. Let DM represent the determinant of B (M : a; b; c)

DM  det[B (M : a; b; c)] De ning D to be 1, it is simple to derive the rst few determinants, thus D = 1 D = b D = b ; ac D = b ; 2abc (B.9) One can also nd the recursion relation DM = bDM ; ; acDM ; (B.10) Eq. B.10 is a linear O
E the solution of which was discussed in Section 4.2. Its characteristic polynomial P (E ) is P (E ; bE + ac) and the two roots to P () = 0 result in the solution 8" p p " #M 9 < b + b ; 4ac #M = 1 b ; b ; 4 ac DM = p ; ; 2 2 b ; 4ac : M = 0; 1; 2;


(B.11) where we have made use of the initial conditions D = 1 and D = b. In the limiting case when b ; 4ac = 0, one can show that 0

0 1

2

2

3

3

1

2

2

+1

2

2

2

0

2

DM = (M + 1) 2b

!M

1

; b = 4ac 2

+1

260

APPENDIX B. SOME PROPERTIES OF TRIDIAGONAL MATRICES

Then for M = 4

B; = 1

and for M = 5

B ;1 =

2 1 666 D4 4

D ;cD cD ;c D ;aD D D ;cD D c D a D ;aD D D D ;cD ;a D a D ;aD D 3

3

2 D4 6 3 1 666 ;a2aD D5 64 ;a3DD21

aD The general element dmn is 4

0

2

2

2

2

1

1

2

0

2

1

1

1

2

1

;cD

3

1

0

2

1

1

1

2

2

3 7 7 7 5

3

3 7 17 7 2 7 7 5 3

cD ;c D c D D D ;cD D c D D ;c D ;aD D D D ;cD D c D a D D ;aD D D D ;cD ;a D aD ;aD D 2

3

1

2

3

1

1

1 3

2

2

1

2

1

3

2

2

2

1

2

2

1

2

1

4

1

3

2

1

0

3

2

1

3

4

Upper triangle:

m = 1; 2;


; M ; 1 ; n = m + 1; m + 2;


; M dmn = Dm; DM ;n(;c)n;m =DM 1

Diagonal:

n = m = 1; 2;


; M dmm = DM ; DM ;m =DM 1

Lower triangle:

m = n + 1; n + 2;


; M ; n = 1; 2;


; M ; 1 dmn = DM ;mDn; (;a)m;n=DM 1

B.4 Eigensystems of Circulant Matrices B.4.1 Standard Tridiagonals

Consider the circulant (see Section 3.4.4) tridiagonal matrix

Bp(M : a; b; c; )

(B.12)

B.4. EIGENSYSTEMS OF CIRCULANT MATRICES

261

The eigenvalues are

 m  ; i(a ; c) sin  2m  ; m = 0; 1; 2;


; M ; 1 m = b + (a + c) cos 2M M (B.13) The right-hand eigenvector that satis es Bp(a; b; c)~xm = m~xm is ~xm = (xj )m = ei j m=M ; j = 0; 1;


; M ; 1 (B.14) p where i  ;1, and the right-hand eigenvector matrix has the form   ij 2m 0; 1;


; M ; 1 X = (xjm) = e M ; mj = = 0; 1;


; M ; 1 The left-hand eigenvector matrix with elements x0 is   2 j ;im M ; m = 0; 1;


; M ; 1 X ; = (x0mj ) = M1 e j = 0; 1;


; M ; 1 Note that both X and X ; are symmetric and that X ; = M1 X , where X  is the conjugate transpose of X . (2

)

1

1

1

B.4.2 General Circulant Systems

Notice the remarkable fact that the elements of the eigenvector matrices X and X ; for the tridiagonal circulant matrix given by eq. B.12 do not depend on the elements a; b; c in the matrix. In fact, all circulant matrices of order M have the same set of linearly independent eigenvectors, even if they are completely dense. An example of a dense circulant matrix of order M = 4 is 2 b b b b 3 6 b b b b 777 6 6 (B.15) 4 b b b b 5 b b b b The eigenvectors are always given by eq. B.14, and further examination shows that the elements in these eigenvectors correspond to the elements in a complex harmonic analysis or complex discrete Fourier series. Although the eigenvectors of a circulant matrix are independent of its elements, the eigenvalues are not. For the element indexing shown in eq. B.15 they have the general form MX ; m = bj ei jm=M 1

0

1

2

3

3

0

1

2

2

3

0

1

1

2

3

0

1

of which eq. B.13 is a special case.

j =0

(2

)

262

APPENDIX B. SOME PROPERTIES OF TRIDIAGONAL MATRICES

B.5 Special Cases Found From Symmetries

Consider a mesh with an even number of interior points such as that shown in Fig. B.1. One can seek from the tridiagonal matrix B (2M : a; b; a; ) the eigenvector subset that has even symmetry when spanning the interval 0  x  . For example, we seek the set of eigenvectors ~xm for which 2 b 6 a 6 6 6 6 6 6 6 4

a b a a ...

32x 3 1 6 7 7 x 6 2 7 7 6 7 ... 77 6 7 6 7 7 6 7 . 7 . 6 7 7 . 7 76 6 a 5 4 x2 75

... a a b a b

x

2 3 x1 6 x2 777 6 6 ... 7 6 6 = m 66 .. 777 .7 6 6 7 4 x2 5

x

1

1

This leads to the subsystem of order M which has the form 2 6 6 6 6 6 6 6 6 6 6 4

3

b a 7 7 a b a 7 7 . . 7 . a ~ ~ ~xm = m~xm 7 B (M : a; b; a)xm = 7 ... a 7 7 a b a 75 a b+a

(B.16)

By folding the known eigenvectors of B (2M : a; b; a) about the center, one can show from previous results that the eigenvalues of eq. B.16 are

m = b + 2a cos (22mM;+1)1

!

; m = 1; 2;


; M

(B.17)

B.6. SPECIAL CASES INVOLVING BOUNDARY CONDITIONS

263

and the corresponding eigenvectors are   ~xm = sin j (2m ; 1) ; 2M + 1 j = 1; 2;


; M

x=

Line of Symmetry

x =0



Imposing symmetry about the same interval but for a mesh with an odd number of points, see Fig. B.1, leads to the matrix 2 3 b a 6 7 a b a 6 7 6 7 . . 6 7 . a B (M : ~a; b; a) = 66 . . . a 777 6 6 4 a b a 75 2a b



~xm = sin j (2m ; 1) 2M

!

     

j0 = 1 2 3 4 5 M0 j= 1 2 3 M b. An odd{numbered mesh Figure B.1 { Symmetrical folds for special cases

By folding the known eigenvalues of B (2M ; 1 : a; b; a) about the center, one can show from previous results that the eigenvalues of eq. B.17 are

; 1) m = b + 2a cos (2m2M and the corresponding eigenvectors are

      

j0 = 1 2 3 4 5 6 M0 j= 1 2 3 M a. An even-numbered mesh Line of Symmetry x =0 x=

!

; m = 1; 2;


; M

; j = 1; 2;


; M

B.6 Special Cases Involving Boundary Conditions We consider two special cases for the matrix operator representing the 3-point central dierence approximation for the second derivative @ =@x at all points away from the boundaries, combined with special conditions imposed at the boundaries. 2

2

APPENDIX B. SOME PROPERTIES OF TRIDIAGONAL MATRICES

264

Note: In both cases

m = 1; 2;


; M j = 1; 2;


; M ;2 + 2 cos() = ;4 sin (=2) 2

When the boundary conditions are Dirichlet on both sides, 2 6 6 6 6 6 6 4

;2

1 1 ;2 1 1 ;2 1 1 ;2 1 1 ;2

3 7 7 7 7 7 7 5



m = ;2 h+2 cos Mim +1 ~xm = sin j m M +1



(B.18)

When one boundary condition is Dirichlet and the other is Neumann (and a diagonal preconditioner is applied to scale the last equation), 2 6 6 6 6 6 6 4

;2

1 1 ;2 1 1 ;2 1 1 ;2 1 1 ;1

3 7 7 7 7 7 7 5

m ~xm



 (2 m ; 1)  = ;2 + 2 cos 2M + 1    (2 m ; 1)  = sin j 2M + 1

(B.19)

Appendix C THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS The Euler equations have a special property that is sometimes useful in constructing numerical methods. In order to examine this property, let us rst inspect Euler's theorem on homogeneous functions. Consider rst the scalar case. If F (u; v) satis es the identity

F (u; v) = nF (u; v) (C.1) for a xed n, F is called homogeneous of degree n. Dierentiating both sides with respect to  and setting  = 1 (since the identity holds for all ), we nd u @F + v @F = nF (u; v) @u @v

(C.2)

F (Q) = nF (Q)

(C.3)

Consider next the theorem as it applies to systems of equations. If the vector F (Q) satis es the identity

for a xed n, F is said to be homogeneous of degree n and we nd "

#

@F Q = nF (Q) @q 265

(C.4)

266APPENDIX C. THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS Now it is easy to show, by direct use of eq. C.3, that both E and F in eqs. 2.11 and 2.12 are homogeneous of degree 1, and their Jacobians, A and B , are homogeneous of degree 0 (actually the latter is a direct consequence of the former).1 This being the case, we notice that the expansion of the ux vector in the vicinity of tn which, according to eq. 6.105 can be written in general as,

E = En + An(Q ; Qn) + O(h2) F = Fn + Bn(Q ; Qn) + O(h2)

(C.5)

can be written

E = An Q + O(h2) F = BnQ + O(h2)

(C.6)

since the terms En ; AnQn and Fn ; BnQn are identically zero for homogeneous vectors of degree 1, see eq. C.4. Notice also that, under this condition, the constant term drops out of eq. 6.106. As a nal remark, we notice from the chain rule that for any vectors F and Q "

#

@F (Q) = @F @Q = A @Q @x @Q @x @x We notice also that for a homogeneous F of degree 1, F = AQ and "

#

@F = A @Q + @A Q @x @x @x

(C.7)

(C.8)

Therefore, if F is homogeneous of degree 1, "

#

@A Q = 0 @x in spite of the fact that individually [@A=@x] and Q are not equal to zero.

(C.9)

Note that this depends on the form of the equation of state. The Euler equations are homogeneous if the equation of state can be written in the form p = f (), where  is the internal energy per unit mass. 1

Bibliography [1] D. Anderson, J. Tannehill, and R. Pletcher. Computational Fluid Mechanics and Heat Transfer. McGraw-Hill Book Company, New York, 1984. [2] C. Hirsch. Numerical Computation of Internal and External Flows, volume 1,2. John Wiley & Sons, New York, 1988.

267