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.

PART I PROBLEMS WITH PERIODIC SOLUTIONS 1 Model equations 3 1.1 Periodic grid functions and difference operators 3 1.2 First–order wave equation, convergence, and stability 9 1.3 Leap–frog scheme 18 1.4 Implicit methods 22 1.5 Truncation error 25 1.6 Heat equation 27 1.7 Convection–diffusion equation 33 1.8 Higher order equations 36 1.9 Second order wave equation 38 1.10 Generalization to several space dimensions 39 2 Higher order accuracy 43 2.1 Efficiency of higher order accurate difference approximations 43 2.2 Time discretization 53 3 Well posed problems 61 3.1 Introduction 61 3.2 Scalar differential equations with constant coefficients in one space dimension 66 3.3 First–order systems with constant coefficients in one space dimension 68 3.4 Parabolic systems with constant coefficients in one space dimension 72 3.5 General systems with constant coefficients 74 3.6 General systems with variable coefficients 75 3.7 Semibounded operators with variable coefficients 77 3.8 Stability and wellposedness 83 3.9 The solution operator and Duhamel’s principle 86 3.10 Generalized solutions 89 3.11 Well–posedness of nonlinear problems 91 3.12 The principle of a priori estimates 93 3.13 The principle of linearization 97 4 Stability and convergence for difference methods 101 4.1 The method of lines 101 4.2 General fully discrete methods 110 4.3 Splitting methods 133 5 Hyperbolic equations and numerical methods 139 5.1 Systems with constant coefficients in one space dimension 139 5.2 Systems with variable coefficients in one space dimension 142 5.3 Systems with constant coefficients in several space dimensions 144 5.4 Systems with variable coefficients in several space dimensions 146 5.5 Approximations with constant coefficients 147 5.6 Approximations with variable coefficients 150 5.7 The method of lines 152 5.8 Staggered grids 156 6 Parabolic equations and numerical methods 163 6.1 General parabolic systems 163 6.2 Stability for difference methods 167 7 Problems with discontinuous solutions 173 7.1 Difference methods for linear hyperbolic problems 173 7.2 Method of characteristics 176 7.3 Method of characteristics in several space dimensions 183 7.4 Method of characteristics on a regular grid 184 7.5 Regularization using viscosity 191 7.6 The inviscid Burgers’ equation 193 7.7 The viscous Burgers’ equation and traveling waves 196 7.8 Numerical methods for scalar equations based on regularization 203 7.9 Regularization for systems of equations 209 7.10 High resolution methods 216 PART II INITIAL–BOUNDARY–VALUE PROBLEMS 8 The energy method for initial–boundary–value problems 227 8.1 Characteristics and boundary conditions for hyperbolic systems in one space dimension 227 8.2 Energy estimates for hyperbolic systems in one space dimension 235 8.3 Energy estimates for parabolic differential equations in one space dimension 242 8.4 Stability and well–posedness for general differential equations 247 8.5 Semibounded operators 250 8.6 Quarter–space problems in more than one space dimension 254 9 The Laplace transformmethod for first order hyperbolic systems261 9.1 A necessary condition for well–posedness 261 9.2 Generalized eigenvalues 264 9.3 The Kreiss condition 266 9.4 Stability in the generalized sense 268 9.5 Derivative boundary conditions for first order hyperbolic systems 275 10 Second order wave equations 279 10.1 The scalar wave equation 279 10.2 General systems of wave equations 294 10.3 A modified wave equation 297 10.4 The elastic wave equations 301 10.5 Einstein’s equations and general relativity 304 11 The energy method for difference approximations 309 11.1 Hyperbolic problems 309 11.2 Parabolic problems 319 11.3 Stability, consistency and order of accuracy 325 11.4 SBP difference operators 330 12 The Laplace transform method for difference approximations 341 12.1 Necessary conditions for stability 341 12.2 Sufficient conditions for stability 350 12.3 Stability in the generalized sense for hyperbolic systems 365 12.4 An example that does not satisfy the Kreiss condition but is stable in the generalized sense 374 12.5 The convergence rate 381 13 The Laplace transform method for fully discrete approximations389 13.1 General theory for approximations of hyperbolic systems 390 13.2 The method of lines and stability in the generalized sense 407 Appendix A: Fourier series and trigonometric interpolation 419 A.1 Some results from the theory of Fourier series 419 A.2 Trigonometric interpolation 423 A.3 Higher dimensions 426 Appendix B: Fourier and Laplace transform 431 B.1 Fourier transform 431 B.2 Laplace transform 434 Appendix C: Some results from linear algebra 437 Appendix D: SBP operators 441 Index 449 References 453

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.