{"product_id":"timedependent-problems-and-difference-methods-9780470900567","title":"TimeDependent Problems and Difference Methods","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003ePraise for the \u003ci\u003eFirst Edition\u003c\/i\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e. . . 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.\u003cbr\u003e \u003ci\u003eSIAM Review\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eTime-Dependent Problems and Difference Methods, Second Edition\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003eThe \u003ci\u003eSecond Edition\u003c\/i\u003e 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 fi\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003c\/p\u003e\u003cp\u003e\u003cb\u003ePreface ix\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePreface to the First Edition xi\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART I PROBLEMS WITH PERIODIC SOLUTIONS 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1. Model Equations 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1. Periodic Gridfunctions and Difference Operators 3\u003c\/p\u003e \u003cp\u003e1.2. First-Order Wave Equation Convergence and Stability 10\u003c\/p\u003e \u003cp\u003e1.3. Leap-Frog Scheme 20\u003c\/p\u003e \u003cp\u003e1.4. Implicit Methods 24\u003c\/p\u003e \u003cp\u003e1.5. Truncation Error 27\u003c\/p\u003e \u003cp\u003e1.6. Heat Equation 30\u003c\/p\u003e \u003cp\u003e1.7. Convection–Diffusion Equation 36\u003c\/p\u003e \u003cp\u003e1.8. Higher Order Equations 39\u003c\/p\u003e \u003cp\u003e1.9. Second-Order Wave Equation 41\u003c\/p\u003e \u003cp\u003e1.10. Generalization to Several Space Dimensions 43\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2. Higher Order Accuracy 47\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1. Efficiency of Higher Order Accurate Difference Approximations 47\u003c\/p\u003e \u003cp\u003e2.2. Time Discretization 57\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3. Well-Posed Problems 65\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1. Introduction 65\u003c\/p\u003e \u003cp\u003e3.2. Scalar Differential Equations with Constant Coefficients in One Space Dimension 70\u003c\/p\u003e \u003cp\u003e3.3. First-Order Systems with Constant Coefficients in One Space Dimension 72\u003c\/p\u003e \u003cp\u003e3.4. Parabolic Systems with Constant Coefficients in One Space Dimension 77\u003c\/p\u003e \u003cp\u003e3.5. General Systems with Constant Coefficients 80\u003c\/p\u003e \u003cp\u003e3.6. General Systems with Variable Coefficients 81\u003c\/p\u003e \u003cp\u003e3.7. Semibounded Operators with Variable Coefficients 83\u003c\/p\u003e \u003cp\u003e3.8. Stability and Well-Posedness 90\u003c\/p\u003e \u003cp\u003e3.9. The Solution Operator and Duhamel’s Principle 93\u003c\/p\u003e \u003cp\u003e3.10. Generalized Solutions 97\u003c\/p\u003e \u003cp\u003e3.11. Well-Posedness of Nonlinear Problems 99\u003c\/p\u003e \u003cp\u003e3.12. The Principle of A Priori Estimates 102\u003c\/p\u003e \u003cp\u003e3.13. The Principle of Linearization 107\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4. Stability and Convergence for Difference Methods 109\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1. The Method of Lines 109\u003c\/p\u003e \u003cp\u003e4.2. General Fully Discrete Methods 119\u003c\/p\u003e \u003cp\u003e4.3. Splitting Methods 147\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5. Hyperbolic Equations and Numerical Methods 153\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1. Systems with Constant Coefficients in One Space Dimension 153\u003c\/p\u003e \u003cp\u003e5.2. Systems with Variable Coefficients in One Space Dimension 156\u003c\/p\u003e \u003cp\u003e5.3. Systems with Constant Coefficients in Several Space Dimensions 158\u003c\/p\u003e \u003cp\u003e5.4. Systems with Variable Coefficients in Several Space Dimensions 160\u003c\/p\u003e \u003cp\u003e5.5. Approximations with Constant Coefficients 162\u003c\/p\u003e \u003cp\u003e5.6. Approximations with Variable Coefficients 165\u003c\/p\u003e \u003cp\u003e5.7. The Method of Lines 167\u003c\/p\u003e \u003cp\u003e5.8. Staggered Grids 172\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6. Parabolic Equations and Numerical Methods 177\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1. General Parabolic Systems 177\u003c\/p\u003e \u003cp\u003e6.2. Stability for Difference Methods 181\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7. Problems with Discontinuous Solutions 189\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1. Difference Methods for Linear Hyperbolic Problems 189\u003c\/p\u003e \u003cp\u003e7.2. Method of Characteristics 193\u003c\/p\u003e \u003cp\u003e7.3. Method of Characteristics in Several Space Dimensions 199\u003c\/p\u003e \u003cp\u003e7.4. Method of Characteristics on a Regular Grid 200\u003c\/p\u003e \u003cp\u003e7.5. Regularization Using Viscosity 208\u003c\/p\u003e \u003cp\u003e7.6. The Inviscid Burgers’ Equation 210\u003c\/p\u003e \u003cp\u003e7.7. The Viscous Burgers’ Equation and Traveling Waves 214\u003c\/p\u003e \u003cp\u003e7.8. Numerical Methods for Scalar Equations Based on Regularization 221\u003c\/p\u003e \u003cp\u003e7.9. Regularization for Systems of Equations 227\u003c\/p\u003e \u003cp\u003e7.10. High Resolution Methods 235\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART II INITIAL–BOUNDARY VALUE PROBLEMS 247\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8. The Energy Method for Initial–Boundary Value Problems 249\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1. Characteristics and Boundary Conditions for Hyperbolic Systems in One Space Dimension 249\u003c\/p\u003e \u003cp\u003e8.2. Energy Estimates for Hyperbolic Systems in One Space Dimension 258\u003c\/p\u003e \u003cp\u003e8.3. Energy Estimates for Parabolic Differential Equations in One Space Dimension 266\u003c\/p\u003e \u003cp\u003e8.4. Stability and Well-Posedness for General Differential Equations 271\u003c\/p\u003e \u003cp\u003e8.5. Semibounded Operators 274\u003c\/p\u003e \u003cp\u003e8.6. Quarter-Space Problems in More than One Space Dimension 279\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9. The Laplace Transform Method for First-Order Hyperbolic Systems 287\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1. A Necessary Condition for Well-Posedness 287\u003c\/p\u003e \u003cp\u003e9.2. Generalized Eigenvalues 291\u003c\/p\u003e \u003cp\u003e9.3. The Kreiss Condition 292\u003c\/p\u003e \u003cp\u003e9.4. Stability in the Generalized Sense 295\u003c\/p\u003e \u003cp\u003e9.5. Derivative Boundary Conditions for First-Order Hyperbolic Systems 303\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10. Second-Order Wave Equations 307\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1. The Scalar Wave Equation 307\u003c\/p\u003e \u003cp\u003e10.2. General Systems of Wave Equations 324\u003c\/p\u003e \u003cp\u003e10.3. A Modified Wave Equation 327\u003c\/p\u003e \u003cp\u003e10.4. The Elastic Wave Equations 331\u003c\/p\u003e \u003cp\u003e10.5. Einstein’s Equations and General Relativity 335\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11. The Energy Method for Difference Approximations 339\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1. Hyperbolic Problems 339\u003c\/p\u003e \u003cp\u003e11.2. Parabolic Problems 350\u003c\/p\u003e \u003cp\u003e11.3. Stability Consistency and Order of Accuracy 357\u003c\/p\u003e \u003cp\u003e11.4. SBP Difference Operators 362\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12. The Laplace Transform Method for Difference Approximations 377\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1. Necessary Conditions for Stability 377\u003c\/p\u003e \u003cp\u003e12.2. Sufficient Conditions for Stability 387\u003c\/p\u003e \u003cp\u003e12.3. Stability in the Generalized Sense for Hyperbolic Systems 405\u003c\/p\u003e \u003cp\u003e12.4. An Example that Does Not Satisfy the Kreiss Condition But is Stable in the Generalized Sense 416\u003c\/p\u003e \u003cp\u003e12.5. The Convergence Rate 423\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13. The Laplace Transform Method for Fully Discrete Approximations 431\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1. General Theory for Approximations of Hyperbolic Systems 431\u003c\/p\u003e \u003cp\u003e13.2. The Method of Lines and Stability in the Generalized Sense 451\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Fourier Series and Trigonometric Interpolation 465\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1. Some Results from the Theory of Fourier Series 465\u003c\/p\u003e \u003cp\u003eA.2. Trigonometric Interpolation 469\u003c\/p\u003e \u003cp\u003eA.3. Higher Dimensions 473\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Fourier and Laplace Transform 477\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1. Fourier Transform 477\u003c\/p\u003e \u003cp\u003eB.2. Laplace Transform 480\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix C Some Results from Linear Algebra 485\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix D SBP Operators 489\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eReferences 499\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eIndex 507\u003c\/b\u003e\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49402453492055,"sku":"9780470900567","price":112.46,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780470900567.jpg?v=1730480447","url":"https:\/\/bookcurl.com\/products\/timedependent-problems-and-difference-methods-9780470900567","provider":"Book Curl","version":"1.0","type":"link"}