Time-Dependent Problems and Difference Methods

**Praise for the __First Edition__** ". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations."—__SIAM Review__ __Time-Dependent Problems and Difference Methods, Second Edition__ continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The __Second Edition__ presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,__Time-Dependent Problems and Difference Methods, Second Edition__ also includes: * High order methods on staggered grids * Extended treatment of Summation By Parts operators and their application to second-order derivatives * Simplified presentation of certain parts and proofs __Time-Dependent Problems and Difference Methods, Second Edition__ is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations. 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 Praise for the First Edition ". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations." — SIAM Review Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations, Time-Dependent Problems and Difference Methods, Second Edition also includes: High order methods on staggered grids Extended treatment of Summation By Parts operators and their application to second-order derivatives Simplified presentation of certain parts and proofs Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations. "Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods. The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations, Time-Dependent Problems and Difference Methods, Second Edition also includes: High order methods on staggered grids ; Extended treatment of Summation By Parts operators and their application to second-order derivatives ; Simplified presentation of certain parts and proofs. Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations."--Publisher's website Written by authors at the forefront of their field, this Second Edition discusses problems with periodic solutions, and presents new information on initial boundary value problems and numerical methods for partial differential equations.