Numerical partial differential equations: 1 Finite difference methods
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1995
|
| Series: | Texts in applied mathematics
22 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 437 S. graph. Darst. |
| ISBN: | 0387979999 |
Staff View
MARC
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| 245 | 1 | 0 | |a Numerical partial differential equations |n 1 |p Finite difference methods |c J. W. Thomas |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Series Preface vii
Preface ix
Contents of Part 2: Conservation Laws and
Elliptic Equations xvii
0 Prelude 1
1 Introduction to Finite Differences 5
1.1 Introduction 5
1.2 Getting Started 6
1.2.1 Implementation 9
1.3 Consistency 10
1.3.1 Special Choice of Ax and At 13
1.4 Neumann Boundary Conditions 14
1.5 Some Variations 16
1.5.1 Lower Order Terms 16
1.5.2 Nonhomogeneous Equations and
Boundary Conditions 17
1.5.3 A Higher Order Scheme 18
1.6 Derivation of Difference Equations 21
1.6.1 Neumann Boundary Conditions 27
1.6.2 Cell Averaged Equations 29
xiv Contents
1.6.3 Cell Centered Grids 32
1.6.4 Nonuniform Grids 34
2 Some Theoretical Considerations 41
2.1 Introduction 41
2.2 Convergence 41
2.2.1 Initial Value Problems 41
2.2.2 Initial Boundary Value Problems 45
2.2.3 A Review of Linear Algebra 48
2.2.4 Some Additional Convergence Topics 54
2.3 Consistency 55
2.3.1 Initial Value Problems 55
2.3.2 Initial Boundary Value Problems 65
2.4 Stability 73
2.4.1 Initial Value Problems 73
2.4.2 Initial Boundary Value Problems 77
2.5 The Lax Theorem 79
2.5.1 Initial Value Problems 79
2.5.2 Initial Boundary Value Problems 81
2.6 Computational Interlude I 83
2.6.1 Review of Computational Results 83
2.6.2 HWO.0.1 84
2.6.3 Implicit Schemes 84
2.6.4 Neumann Boundary Conditions 89
2.6.5 Derivation of Implicit Schemes 93
3 Stability 97
3.1 Analysis of Stability 97
3.1.1 Initial Value Problems 97
3.1.2 Initial Boundary Value Problems 112
3.2 Finite Fourier Series and Stability 117
3.3 Gerschgorin Circle Theorem 132
3.4 Computational Interlude II 137
3.4.1 Review of Computational Results 137
3.4.2 HWO.0.1 139
4 Parabolic Equations 147
4.1 Introduction 147
4.2 Two Dimensional Parabolic Equations 148
4.2.1 Neumann Boundary Conditions 151
4.2.2 Derivation of Difference Equations 153
4.3 Convergence, Consistency, Stability 156
4.3.1 Stability of Initial Value Schemes 157
4.3.2 Stability of Initial Boundary Value Schemes .... 160
4.4 Alternating Direction Implicit Schemes 164
Contents xv
4.4.1 Peaceman Rachford Scheme 165
4.4.2 Initial Value Problems 165
4.4.3 Initial Boundary Value Problems 169
4.4.4 Douglas Rachford Scheme 183
4.4.5 Nonhomogeneous ADI Schemes 185
4.4.6 Three Dimensional Schemes 190
4.5 Polar Coordinates 193
5 Hyperbolic Equations 205
5.1 Introduction 205
5.2 Initial Value Problems 206
5.3 Numerical Solution of Initial Value Problems 209
5.3.1 One Sided Schemes 210
5.3.2 Centered Scheme 213
5.3.3 Lax Wendroff Scheme 213
5.3.4 More Explicit Schemes 216
5.4 Implicit Schemes 216
5.4.1 One Sided Schemes 216
5.4.2 Centered Scheme 217
5.4.3 Lax Wendroff Scheme 218
5.4.4 Crank Nicolson Scheme 219
5.5 Initial Boundary Value Problems 220
5.5.1 Periodic Boundary Conditions 221
5.5.2 Dirichlet Boundary Conditions 221
5.6 Numerical Solution of Initial Boundary Value Problems . . 223
5.6.1 Periodic Boundary Conditions 223
5.6.2 Dirichlet Boundary Conditions 229
5.7 The Courant Friedrichs Lewy Condition 232
5.8 Two Dimensional Hyperbolic Equations 239
5.8.1 Conservation Law Derivation 239
5.8.2 Initial Value Problems 242
5.8.3 ADI Schemes 247
5.8.4 Courant Friedrichs Lewy Condition for
Two Dimensional Problems 249
5.8.5 Two Dimensional Initial Boundary Value Problems 251
5.9 Computational Interlude III 254
5.9.1 Review of Computational Results 254
5.9.2 Convection Diffusion Equations 256
5.9.3 HW0.0.1 257
5.9.4 HWO.0.2 258
6 Systems of Partial Differential Equations 261
6.1 Introduction 261
6.2 Initial Value Difference Schemes 263
xvi Contents
6.2.1 Flux Splitting 278
6.2.2 Implicit Schemes 280
6.3 Initial Boundary Value Problems 284
6.3.1 Boundary Conditions 285
6.3.2 Implementation 292
6.4 Multilevel Schemes 300
6.4.1 Scalar Multilevel Schemes 300
6.4.2 Implementation of Scalar Multilevel Schemes .... 306
6.4.3 Multilevel Systems 310
6.5 Higher Order Hyperbolic Equations 312
6.5.1 Initial Value Problems 314
6.5.2 More 320
6.6 Courant Friedrichs Lewy Condition for Systems 322
6.7 Two Dimensional Systems 324
6.7.1 Initial Value Problems 325
6.7.2 Boundary Conditions 333
6.7.3 Two Dimensional Multilevel Schemes 338
6.8 Simultaneously Diagonalizable Matrices? 340
6.9 A Consistent, Convergent, Unstable Difference Scheme? . . 342
6.10 Computational Interlude IV 344
6.10.1 HW0.0.1 and HW0.0.2 344
6.10.2 HWO.0.3 347
6.10.3 Parabolic Problems in Polar Coordinates 351
6.10.4 An Alternate Scheme for Polar Coordinates 356
7 Dispersion and Dissipation 361
7.1 Introduction 361
7.1.1 HW5.6.3 361
7.1.2 HW5.6.5 365
7.2 Dispersion and Dissipation for Partial
Differential Equations 365
7.3 Dispersion and Dissipation for Difference Equations .... 369
7.4 Dispersion Analysis for the Leapfrog Scheme 388
7.5 More Dissipation 396
7.6 Artificial Dissipation 399
7.7 Modified Partial Differential Equation 404
7.8 Discontinuous Solutions 413
7.9 Computational Interlude V 422
7.9.1 HW0.0.1 423
7.9.2 HWO.0.3 425
References 427
Index 429
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| indexdate | 2024-07-09T18:30:25Z |
| institution | BVB |
| isbn | 0387979999 |
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| publishDate | 1995 |
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| publisher | Springer |
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| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Thomas, James W. 1941- Verfasser (DE-588)121186881 aut Numerical partial differential equations 1 Finite difference methods J. W. Thomas New York [u.a.] Springer 1995 XX, 437 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 22 Texts in applied mathematics ... (DE-604)BV012602928 1 Texts in applied mathematics 22 (DE-604)BV002476038 22 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008558822&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 1 Finite difference methods J. W. Thomas |
| title_fullStr | Numerical partial differential equations 1 Finite difference methods J. W. Thomas |
| title_full_unstemmed | Numerical partial differential equations 1 Finite difference methods J. W. Thomas |
| title_short | Numerical partial differential equations |
| title_sort | numerical partial differential equations finite difference methods |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008558822&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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