Time-Dependent Problems and Difference Methods

TIME-DEPENDENT PROBLEMS AND DIFFERENCE METHODS......Page 3
CONTENTS......Page 7
Preface......Page 11
Preface to the First Edition......Page 13
PART I PROBLEMS WITH PERIODIC SOLUTIONS......Page 17
1.1. Periodic Gridfunctions and Difference Operators......Page 19
1.2. First-Order Wave Equation, Convergence, and Stability......Page 26
1.3. Leap-Frog Scheme......Page 36
1.4. Implicit Methods......Page 40
1.5. Truncation Error......Page 43
1.6. Heat Equation......Page 46
1.7. Convection–Diffusion Equation......Page 52
1.8. Higher Order Equations......Page 55
1.9. Second-Order Wave Equation......Page 57
1.10. Generalization to Several Space Dimensions......Page 59
2.1. Efficiency of Higher Order Accurate Difference Approximations......Page 63
2.2. Time Discretization......Page 73
3.1. Introduction......Page 81
3.2. Scalar Differential Equations with Constant Coefficients in One Space Dimension......Page 86
3.3. First-Order Systems with Constant Coefficients in One Space Dimension......Page 88
3.4. Parabolic Systems with Constant Coefficients in One Space Dimension......Page 93
3.5. General Systems with Constant Coefficients......Page 96
3.6. General Systems with Variable Coefficients......Page 97
3.7. Semibounded Operators with Variable Coefficients......Page 99
3.8. Stability and Well-Posedness......Page 106
3.9. The Solution Operator and Duhamel\'s Principle......Page 109
3.10. Generalized Solutions......Page 113
3.11. Well-Posedness of Nonlinear Problems......Page 115
3.12. The Principle of A Priori Estimates......Page 118
3.13. The Principle of Linearization......Page 123
4.1. The Method of Lines......Page 125
4.2. General Fully Discrete Methods......Page 135
4.3. Splitting Methods......Page 163
5.1. Systems with Constant Coefficients in One Space Dimension......Page 169
5.2. Systems with Variable Coefficients in One Space Dimension......Page 172
5.3. Systems with Constant Coefficients in Several Space Dimensions......Page 174
5.4. Systems with Variable Coefficients in Several Space Dimensions......Page 176
5.5. Approximations with Constant Coefficients......Page 178
5.6. Approximations with Variable Coefficients......Page 181
5.7. The Method of Lines......Page 183
5.8. Staggered Grids......Page 188
6.1. General Parabolic Systems......Page 193
6.2. Stability for Difference Methods......Page 197
7.1. Difference Methods for Linear Hyperbolic Problems......Page 205
7.2. Method of Characteristics......Page 209
7.3. Method of Characteristics in Several Space Dimensions......Page 215
7.4. Method of Characteristics on a Regular Grid......Page 216
7.5. Regularization Using Viscosity......Page 224
7.6. The Inviscid Burgers\' Equation......Page 226
7.7. The Viscous Burgers\' Equation and Traveling Waves......Page 230
7.8. Numerical Methods for Scalar Equations Based on Regularization......Page 237
7.9. Regularization for Systems of Equations......Page 243
7.10. High Resolution Methods......Page 251
PART II INITIAL–BOUNDARY VALUE PROBLEMS......Page 263
8.1. Characteristics and Boundary Conditions for Hyperbolic Systems in One Space Dimension......Page 265
8.2. Energy Estimates for Hyperbolic Systems in One Space Dimension......Page 274
8.3. Energy Estimates for Parabolic Differential Equations in One Space Dimension......Page 282
8.4. Stability and Well-Posedness for General Differential Equations......Page 287
8.5. Semibounded Operators......Page 290
8.6. Quarter-Space Problems in More than One Space Dimension......Page 295
9.1. A Necessary Condition for Well-Posedness......Page 303
9.2. Generalized Eigenvalues......Page 307
9.3. The Kreiss Condition......Page 308
9.4. Stability in the Generalized Sense......Page 311
9.5. Derivative Boundary Conditions for First-Order Hyperbolic Systems......Page 319
10.1. The Scalar Wave Equation......Page 323
10.2. General Systems of Wave Equations......Page 340
10.3. A Modified Wave Equation......Page 343
10.4. The Elastic Wave Equations......Page 347
10.5. Einstein\'s Equations and General Relativity......Page 351
11.1. Hyperbolic Problems......Page 355
11.2. Parabolic Problems......Page 366
11.3. Stability, Consistency, and Order of Accuracy......Page 373
11.4. SBP Difference Operators......Page 378
12.1. Necessary Conditions for Stability......Page 393
12.2. Sufficient Conditions for Stability......Page 403
12.3. Stability in the Generalized Sense for Hyperbolic Systems......Page 421
12.4. An Example that Does Not Satisfy the Kreiss Condition But is Stable in the Generalized Sense......Page 432
12.5. The Convergence Rate......Page 439
13.1. General Theory for Approximations of Hyperbolic Systems......Page 447
13.2. The Method of Lines and Stability in the Generalized Sense......Page 467
A.1. Some Results from the Theory of Fourier Series......Page 481
A.2. Trigonometric Interpolation......Page 485
A.3. Higher Dimensions......Page 489
B.1. Fourier Transform......Page 493
B.2. Laplace Transform......Page 496
Appendix C Some Results from Linear Algebra......Page 501
Appendix D SBP Operators......Page 505
References......Page 515
Index......Page 523