Time-dependent problems and difference methods
- Bertil Gustafsson, Heinz-Otto Kreiss, Joseph Oliger.
- Second edition.
- Hoboken, New Jersey : Wiley, 
- Physical description
- xiii, 509 pages : illustrations ; 25 cm.
- Pure and applied mathematics (John Wiley & Sons : Unnumbered)
Math & Statistics Library
QA374 .G974 2013
- Unknown QA374 .G974 2013
- Includes bibliographical references (pages 499-505) and index.
- 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 DifferenceApproximations 47 2.2. Time Discretization 57 3. Well-Posed Problems 65 3.1. Introduction 65 3.2. Scalar Differential Equations with Constant Coefficients inOne Space Dimension 70 3.3. First-Order Systems with Constant Coefficients in One SpaceDimension 72 3.4. Parabolic Systems with Constant Coefficients in One SpaceDimension 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 Methods109 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 Dimension153 5.2. Systems with Variable Coefficients in One Space Dimension156 5.3. Systems with Constant Coefficients in Several SpaceDimensions 158 5.4. Systems with Variable Coefficients in Several SpaceDimensions 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 Dimensions199 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 Waves214 7.8. Numerical Methods for Scalar Equations Based onRegularization 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 ValueProblems 249 8.1. Characteristics and Boundary Conditions for HyperbolicSystems in One Space Dimension 249 8.2. Energy Estimates for Hyperbolic Systems in One SpaceDimension 258 8.3. Energy Estimates for Parabolic Differential Equations inOne Space Dimension 266 8.4. Stability and Well-Posedness for General DifferentialEquations 271 8.5. Semibounded Operators 274 8.6. Quarter-Space Problems in More than One Space Dimension279 9. The Laplace Transform Method for First-Order HyperbolicSystems 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 HyperbolicSystems 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 Approximations339 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 DifferenceApproximations 377 12.1. Necessary Conditions for Stability 377 12.2. Sufficient Conditions for Stability 387 12.3. Stability in the Generalized Sense for Hyperbolic Systems405 12.4. An Example that Does Not Satisfy the Kreiss Condition Butis Stable in the Generalized Sense 416 12.5. The Convergence Rate 423 13. The Laplace Transform Method for Fully DiscreteApproximations 431 13.1. General Theory for Approximations of Hyperbolic Systems431 13.2. The Method of Lines and Stability in the Generalized Sense451 Appendix A Fourier Series and Trigonometric Interpolation465 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.
- (source: Nielsen Book Data)
- Publisher's Summary
- Praise for the First Edition "...fills a considerable gap in the numerical analysisliterature by providing a self-contained treatment ...this is animportant work written in a clear style ...warmly recommended toany graduate student or researcher in the field of the numericalsolution of partial differential equations." SIAM Review Time-Dependent Problems and Difference Methods, SecondEdition continues to provide guidance for the analysis ofdifference methods for computing approximate solutions to partialdifferential equations for time-dependent problems. The book treatsdifferential equations and difference methods with a paralleldevelopment, thus achieving a more useful analysis of numericalmethods. The Second Edition presents hyperbolic equations in greatdetail as well as new coverage on second-order systems of waveequations including acoustic waves, elastic waves, and Einsteinequations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain newproperties that must be taken into account when analyzingstability. Featuring the latest material in partial differentialequations with new theorems, examples, andillustrations, Time-Dependent Problems and Difference Methods, Second Edition also includes: * High order methods on staggered grids * Extended treatment of Summation By Parts operators and theirapplication to second-order derivatives * Simplified presentation of certain parts and proofs Time-Dependent Problems and Difference Methods, SecondEdition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who usenumerical experiments to test designs and to predict andinvestigate physical phenomena. The book is also excellent forgraduate-level courses in applied mathematics and scientificcomputations.
(source: Nielsen Book Data)
- Publication date
- Pure and applied mathematics
- "Published simultaneously in Canada"--Title page verso.
- 9780470900567 (cloth)
- 0470900563 (cloth)