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 ixPreface to the First Edition xiPART I PROBLEMS WITH PERIODIC SOLUTIONS 11. Model Equations 31.1. Periodic Gridfunctions and Difference Operators 31.2. First-Order Wave Equation Convergence and Stability 101.3. Leap-Frog Scheme 201.4. Implicit Methods 241.5. Truncation Error 271.6. Heat Equation 301.7. Convection–Diffusion Equation 361.8. Higher Order Equations 391.9. Second-Order Wave Equation 411.10. Generalization to Several Space Dimensions 432. Higher Order Accuracy 472.1. Efficiency of Higher Order Accurate Difference Approximations 472.2. Time Discretization 573. Well-Posed Problems 653.1. Introduction 653.2. Scalar Differential Equations with Constant Coefficients in One Space Dimension 703.3. First-Order Systems with Constant Coefficients in One Space Dimension 723.4. Parabolic Systems with Constant Coefficients in One Space Dimension 773.5. General Systems with Constant Coefficients 803.6. General Systems with Variable Coefficients 813.7. Semibounded Operators with Variable Coefficients 833.8. Stability and Well-Posedness 903.9. The Solution Operator and Duhamel’s Principle 933.10. Generalized Solutions 973.11. Well-Posedness of Nonlinear Problems 993.12. The Principle of A Priori Estimates 1023.13. The Principle of Linearization 1074. Stability and Convergence for Difference Methods 1094.1. The Method of Lines 1094.2. General Fully Discrete Methods 1194.3. Splitting Methods 1475. Hyperbolic Equations and Numerical Methods 1535.1. Systems with Constant Coefficients in One Space Dimension 1535.2. Systems with Variable Coefficients in One Space Dimension 1565.3. Systems with Constant Coefficients in Several Space Dimensions 1585.4. Systems with Variable Coefficients in Several Space Dimensions 1605.5. Approximations with Constant Coefficients 1625.6. Approximations with Variable Coefficients 1655.7. The Method of Lines 1675.8. Staggered Grids 1726. Parabolic Equations and Numerical Methods 1776.1. General Parabolic Systems 1776.2. Stability for Difference Methods 1817. Problems with Discontinuous Solutions 1897.1. Difference Methods for Linear Hyperbolic Problems 1897.2. Method of Characteristics 1937.3. Method of Characteristics in Several Space Dimensions 1997.4. Method of Characteristics on a Regular Grid 2007.5. Regularization Using Viscosity 2087.6. The Inviscid Burgers’ Equation 2107.7. The Viscous Burgers’ Equation and Traveling Waves 2147.8. Numerical Methods for Scalar Equations Based on Regularization 2217.9. Regularization for Systems of Equations 2277.10. High Resolution Methods 235PART II INITIAL–BOUNDARY VALUE PROBLEMS 2478. The Energy Method for Initial–Boundary Value Problems 2498.1. Characteristics and Boundary Conditions for Hyperbolic Systems in One Space Dimension 2498.2. Energy Estimates for Hyperbolic Systems in One Space Dimension 2588.3. Energy Estimates for Parabolic Differential Equations in One Space Dimension 2668.4. Stability and Well-Posedness for General Differential Equations 2718.5. Semibounded Operators 2748.6. Quarter-Space Problems in More than One Space Dimension 2799. The Laplace Transform Method for First-Order Hyperbolic Systems 2879.1. A Necessary Condition for Well-Posedness 2879.2. Generalized Eigenvalues 2919.3. The Kreiss Condition 2929.4. Stability in the Generalized Sense 2959.5. Derivative Boundary Conditions for First-Order Hyperbolic Systems 30310. Second-Order Wave Equations 30710.1. The Scalar Wave Equation 30710.2. General Systems of Wave Equations 32410.3. A Modified Wave Equation 32710.4. The Elastic Wave Equations 33110.5. Einstein’s Equations and General Relativity 33511. The Energy Method for Difference Approximations 33911.1. Hyperbolic Problems 33911.2. Parabolic Problems 35011.3. Stability Consistency and Order of Accuracy 35711.4. SBP Difference Operators 36212. The Laplace Transform Method for Difference Approximations 37712.1. Necessary Conditions for Stability 37712.2. Sufficient Conditions for Stability 38712.3. Stability in the Generalized Sense for Hyperbolic Systems 40512.4. An Example that Does Not Satisfy the Kreiss Condition But is Stable in the Generalized Sense 41612.5. The Convergence Rate 42313. The Laplace Transform Method for Fully Discrete Approximations 43113.1. General Theory for Approximations of Hyperbolic Systems 43113.2. The Method of Lines and Stability in the Generalized Sense 451Appendix A Fourier Series and Trigonometric Interpolation 465A.1. Some Results from the Theory of Fourier Series 465A.2. Trigonometric Interpolation 469A.3. Higher Dimensions 473Appendix B Fourier and Laplace Transform 477B.1. Fourier Transform 477B.2. Laplace Transform 480Appendix C Some Results from Linear Algebra 485Appendix D SBP Operators 489References 499Index 507