Time-Dependent Problems and Difference Methods

BERTIL GUSTAFSSON, PhD, is Professor Emeritus in the Department of Information Technology at Uppsala University and is well known for his work in initial-boundary value problems. HEINZ-OTTO KREISS, PhD, is Professor Emeritus in the Department of Mathematics at University of California, Los Angeles and is a renowned mathematician in the field of applied mathematics. JOSEPH OLIGER, PhD, was Professor in the Department of Computer Science at Stanford University and was well known for his early research in numerical methods for partial differential equations.

Preface ix

Preface to the First Edition xi

PART I PROBLEMS WITH PERIODIC SOLUTIONS 1

1. Model Equations 3

1.1. Periodic Gridfunctions and Difference Operators 3

1.2. First-Order Wave Equation Convergence and Stability 10

1.3. Leap-Frog Scheme 20

1.4. Implicit Methods 24

1.5. Truncation Error 27

1.6. Heat Equation 30

1.7. Convection–Diffusion Equation 36

1.8. Higher Order Equations 39

1.9. Second-Order Wave Equation 41

1.10. Generalization to Several Space Dimensions 43

2. Higher Order Accuracy 47

2.1. Efficiency of Higher Order Accurate Difference Approximations 47

2.2. Time Discretization 57

3. Well-Posed Problems 65

3.1. Introduction 65

3.2. Scalar Differential Equations with Constant Coefficients in One Space Dimension 70

3.3. First-Order Systems with Constant Coefficients in One Space Dimension 72

3.4. Parabolic Systems with Constant Coefficients in One Space Dimension 77

3.5. General Systems with Constant Coefficients 80

3.6. General Systems with Variable Coefficients 81

3.7. Semibounded Operators with Variable Coefficients 83

3.8. Stability and Well-Posedness 90

3.9. The Solution Operator and Duhamel’s Principle 93

3.10. Generalized Solutions 97

3.11. Well-Posedness of Nonlinear Problems 99

3.12. The Principle of A Priori Estimates 102

3.13. The Principle of Linearization 107

4. Stability and Convergence for Difference Methods 109

4.1. The Method of Lines 109

4.2. General Fully Discrete Methods 119

4.3. Splitting Methods 147

5. Hyperbolic Equations and Numerical Methods 153

5.1. Systems with Constant Coefficients in One Space Dimension 153

5.2. Systems with Variable Coefficients in One Space Dimension 156

5.3. Systems with Constant Coefficients in Several Space Dimensions 158

5.4. Systems with Variable Coefficients in Several Space Dimensions 160

5.5. Approximations with Constant Coefficients 162

5.6. Approximations with Variable Coefficients 165

5.7. The Method of Lines 167

5.8. Staggered Grids 172

6. Parabolic Equations and Numerical Methods 177

6.1. General Parabolic Systems 177

6.2. Stability for Difference Methods 181

7. Problems with Discontinuous Solutions 189

7.1. Difference Methods for Linear Hyperbolic Problems 189

7.2. Method of Characteristics 193

7.3. Method of Characteristics in Several Space Dimensions 199

7.4. Method of Characteristics on a Regular Grid 200

7.5. Regularization Using Viscosity 208

7.6. The Inviscid Burgers’ Equation 210

7.7. The Viscous Burgers’ Equation and Traveling Waves 214

7.8. Numerical Methods for Scalar Equations Based on Regularization 221

7.9. Regularization for Systems of Equations 227

7.10. High Resolution Methods 235

PART II INITIAL–BOUNDARY VALUE PROBLEMS 247

8. The Energy Method for Initial–Boundary Value Problems 249

8.1. Characteristics and Boundary Conditions for Hyperbolic Systems in One Space Dimension 249

8.2. Energy Estimates for Hyperbolic Systems in One Space Dimension 258

8.3. Energy Estimates for Parabolic Differential Equations in One Space Dimension 266

8.4. Stability and Well-Posedness for General Differential Equations 271

8.5. Semibounded Operators 274

8.6. Quarter-Space Problems in More than One Space Dimension 279

9. The Laplace Transform Method for First-Order Hyperbolic Systems 287

9.1. A Necessary Condition for Well-Posedness 287

9.2. Generalized Eigenvalues 291

9.3. The Kreiss Condition 292

9.4. Stability in the Generalized Sense 295

9.5. Derivative Boundary Conditions for First-Order Hyperbolic Systems 303

10. Second-Order Wave Equations 307

10.1. The Scalar Wave Equation 307

10.2. General Systems of Wave Equations 324

10.3. A Modified Wave Equation 327

10.4. The Elastic Wave Equations 331

10.5. Einstein’s Equations and General Relativity 335

11. The Energy Method for Difference Approximations 339

11.1. Hyperbolic Problems 339

11.2. Parabolic Problems 350

11.3. Stability Consistency and Order of Accuracy 357

11.4. SBP Difference Operators 362

12. The Laplace Transform Method for Difference Approximations 377

12.1. Necessary Conditions for Stability 377

12.2. Sufficient Conditions for Stability 387

12.3. Stability in the Generalized Sense for Hyperbolic Systems 405

12.4. An Example that Does Not Satisfy the Kreiss Condition But is Stable in the Generalized Sense 416

12.5. The Convergence Rate 423

13. The Laplace Transform Method for Fully Discrete Approximations 431

13.1. General Theory for Approximations of Hyperbolic Systems 431

13.2. The Method of Lines and Stability in the Generalized Sense 451

Appendix A Fourier Series and Trigonometric Interpolation 465

A.1. Some Results from the Theory of Fourier Series 465

A.2. Trigonometric Interpolation 469

A.3. Higher Dimensions 473

Appendix B Fourier and Laplace Transform 477

B.1. Fourier Transform 477

B.2. Laplace Transform 480

Appendix C Some Results from Linear Algebra 485

Appendix D SBP Operators 489

References 499

Index 507