Numerical partial differential equations: 2 Conservation laws and elliptic equations
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1999
|
| Schriftenreihe: | Texts in applied mathematics
33 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXII, 556 S. Ill., graph. Darst. |
| ISBN: | 0387983465 |
Internformat
MARC
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| 100 | 1 | |a Thomas, James W. |d 1941- |e Verfasser |0 (DE-588)121186881 |4 aut | |
| 245 | 1 | 0 | |a Numerical partial differential equations |n 2 |p Conservation laws and elliptic equations |c J. W. Thomas |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XXII, 556 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
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| adam_text | Contents
Series Preface vii
Preface ix
Preface to Part 1 xi
Contents of Part 1 xix
8 Stability of Initial Boundary Value Schemes 1
8.1 Introduction 1
8.2 Stability 2
8.2.1 Stability: An Easy Case 3
8.2.2 Stability: Another Easy Case 24
8.2.3 GKSO: General Theory 39
8.2.4 Left Quarter Plane Problems 47
8.3 Constructing Stable Difference Schemes 51
8.4 Consistency and Convergence 55
8.4.1 Norms and Consistency 56
8.4.2 Consistency of Numerical Boundary Conditions . . 57
8.4.3 Convergence Theorem: Gustafsson 59
8.5 Schemes Without Numerical Boundary Conditions 62
8.6 Parabolic Initial Boundary Value Problems 64
xvi Contents
9 Conservation Laws 73
9.1 Introduction 73
9.2 Theory of Scalar Conservation Laws 75
9.2.1 Shock Formation 76
9.2.2 Weak Solutions 81
9.2.3 Discontinuous Solutions 88
9.2.4 The Entropy Condition 97
9.2.5 Solution of Scalar Conservation Laws 105
9.3 Theory of Systems of Conservation Laws 113
9.3.1 Solutions of Riemann Problems 120
9.4 Computational Interlude VI 134
9.5 Numerical Solution of Conservation Laws 140
9.5.1 Introduction 140
9.6 Difference Schemes for Conservation Laws 150
9.6.1 Consistency 151
9.6.2 Conservative Schemes 154
9.6.3 Discrete Conservation 161
9.6.4 The Courant Friedrichs Lewy Condition 162
9.6.5 Entropy 164
9.7 Difference Schemes for Scalar Conservation Laws 169
9.7.1 Definitions 169
9.7.2 Theorems 176
9.7.3 Godunov Scheme 194
9.7.4 High Resolution Schemes 204
9.7.5 Flux Limiter Methods 205
9.7.6 Slope Limiter Methods 221
9.7.7 Modified Flux Method 229
9.8 Difference Schemes for if System Conservation Laws . . . 236
9.9 Godunov Schemes 236
9.9.1 Godunov Schemes for Linear if System
Conservation Laws 236
9.9.2 Godunov Schemes for i^ System Conservation Laws 238
9.9.3 Approximate Riemann Solvers: Theory 241
9.9.4 Approximate Riemann Solvers: Applications .... 245
9.10 High Resolution Schemes for Linear if System
Conservation Laws 259
9.10.1 Flux Limiter Schemes for Linear if System
Conservation Laws 260
9.10.2 Slope Limiter Schemes for Linear if System
Conservation Laws 262
9.10.3 A Modified Flux Scheme for Linear if System
Conservation Laws 263
9.10.4 High Resolution Schemes for if System
Conservation Laws 265
9.11 Implicit Schemes 266
Contents xvii
9.12 Difference Schemes for Two Dimensional Conservation Laws 269
9.12.1 Some Computational Examples 277
9.12.2 Some Two Dimensional High Resolution Schemes . 278
9.12.3 The Zalesak Smolarkiewicz Scheme 284
9.12.4 A Z S Scheme for Nonlinear Conservation Laws . . 290
9.12.5 Two Dimensional if System Conservation Laws . . 292
10 Elliptic Equations 295
10.1 Introduction 295
10.2 Solvability of Elliptic Difference Equations: Dirichlet
Boundary Conditions 297
10.3 Convergence of Elliptic Difference Schemes: Dirichlet
Boundary Conditions 303
10.4 Solution Schemes for Elliptic Difference Equations:
Introduction 308
10.5 Residual Correction Methods 308
10.5.1 Analysis of Residual Correction Schemes 310
10.5.2 Jacobi Relaxation Scheme 312
10.5.3 Analysis of the Jacobi Relaxation Scheme 315
10.5.4 Stopping Criteria 319
10.5.5 Implementation of the Jacobi Scheme 326
10.5.6 Gauss Seidel Scheme 328
10.5.7 Analysis of the Gauss Seidel Relaxation Scheme . . 332
10.5.8 Successive Overrelaxation Scheme 335
10.5.9 Elementary Analysis of SOR Scheme 336
10.5.10 More on the SOR Scheme 354
10.5.11 Line Jacobi, Gauss Seidel and SOR Schemes .... 360
10.5.12 Approximating Wb . Reality 368
10.6 Elliptic Difference Equations: Neumann
Boundary Conditions 371
10.6.1 First Order Approximation 372
10.6.2 Second Order Approximation 379
10.6.3 Second Order Approximation on an Offset Grid . . 384
10.7 Numerical Solution of Neumann Problems 386
10.7.1 Introduction 386
10.7.2 Residual Correction Schemes 387
10.7.3 Jacobi and Gauss Seidel Iteration 388
10.7.4 SOR Scheme 392
10.7.5 Approximation of Wf, 392
10.7.6 Implementation: Neumann Problems 394
10.8 Elliptic Difference Equations: Mixed Problems 396
10.8.1 Introduction 396
10.8.2 Mixed Problems: Solvability 401
10.8.3 Mixed Problems: Implementation 404
10.9 Elliptic Difference Equations: Polar Coordinates 406
xviii Contents
10.10 Multigrid 412
10.10.1 Introduction 412
10.10.2 Smoothers 415
10.10.3 Grid Transfers 420
10.10.4 Multigrid Algorithm 425
10.11 Computational Interlude VII 448
10.11.1 Blocking Out: Irregular Regions 448
10.11.2 HW0.0.4 457
10.12 ADI Schemes 460
10.13 Conjugate Gradient Scheme 466
10.13.1 Preconditioned Conjugate Gradient Scheme .... 471
10.13.2 SSOR as a Preconditioner 475
10.13.3 Implementation 476
10.14 Using Iterative Methods to Solve Time Dependent Problems 479
10.15 Using FFTs to Solve Elliptic Problems 481
10.16 Computational Interlude VIII 488
11 Irregular Regions and Grids 493
11.1 Introduction 493
11.2 Irregular Geometries 493
11.2.1 Blocking Out 493
11.2.2 Map the Region 498
11.2.3 Grid Generation 502
11.3 Grid Refinement 514
11.3.1 Grid Refinement: Explicit Schemes for Hyperbolic
Problems 523
11.3.2 Grid Refinement for Implicit Schemes 525
11.4 Unstructured Grids 530
References 535
Index 541
|
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| id | DE-604.BV012602946 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:30:25Z |
| institution | BVB |
| isbn | 0387983465 |
| language | English |
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| physical | XXII, 556 S. Ill., graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Thomas, James W. 1941- Verfasser (DE-588)121186881 aut Numerical partial differential equations 2 Conservation laws and elliptic equations J. W. Thomas New York [u.a.] Springer 1999 XXII, 556 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 33 Texts in applied mathematics ... (DE-604)BV012602928 2 Texts in applied mathematics 33 (DE-604)BV002476038 33 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008558832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Thomas, James W. 1941- Numerical partial differential equations Texts in applied mathematics |
| title | Numerical partial differential equations |
| title_auth | Numerical partial differential equations |
| title_exact_search | Numerical partial differential equations |
| title_full | Numerical partial differential equations 2 Conservation laws and elliptic equations J. W. Thomas |
| title_fullStr | Numerical partial differential equations 2 Conservation laws and elliptic equations J. W. Thomas |
| title_full_unstemmed | Numerical partial differential equations 2 Conservation laws and elliptic equations J. W. Thomas |
| title_short | Numerical partial differential equations |
| title_sort | numerical partial differential equations conservation laws and elliptic equations |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008558832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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