Numerical methods for wave equations in geophysical fluid dynamics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1999
|
| Schriftenreihe: | Texts in applied mathematics
32 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 443 - 455 |
| Beschreibung: | XVII, 465 S. graph. Darst. |
| ISBN: | 0387983767 |
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Datensatz im Suchindex
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| adam_text | IMAGE 1
DALE R. DURRAN
NUMERICAL METHODS FOR WAVE EQUATIONS IN GEOPHYSICAL FLUID DYNAMICS
WITH 93 ILLUSTRATIONS
SPRINGER
IMAGE 2
CONTENTS
SERIES PREFACE VII
PREFACE IX
1 INTRODUCTION 1
1.1 PARTIAL DIFFERENTIAL EQUATIONS-SOME BASICS 2
1.1.1 FIRST-ORDER HYPERBOLIC EQUATIONS 4
1.1.2 LINEAR SECOND-ORDER EQUATIONS IN TWO INDEPENDENT VARIABLES 7
1.2 WAVC EQUATIONS IN DCOPHYSICAL FLUID DYNAMICS 11
1.2.1 HYPERBOLIC EQUATIONS 12
1.2.2 FILTCRCD EQUATIONS 20
1.3 STRATCGIES FOR NUMCRICAL APPROXIMATION 26
1.3.1 APPROXIMALING CALCULUS WITH ALPCBRA 26
1.3.2 MARCHING SCHEMAS 29
PROBLEMS 33
2 BASIC FINITE-DIFLERCNCC METHODS 35
2.1 ACCURACY AND CONSISTCNCY 36
2.2 STABILITY AND CONVCRGCNCC 39
2.2.1 THE ENERGY METHOD 41
2.2.2 VON NEUMANN S METHOD 43
2.2.3 THE COURANT- FRCDRICHS-1-EWY CONDITION 45
2.3 TIMC-DIFFERENCING 47
2.3.1 THE OSCILLATION EQUATION: PHASE-SPEED AND AMPLITUDE ERROR 48
IMAGE 3
XIV CONTENTS
2.3.2 SINGLE-STAGE TWO-LEVEL SCHEMES 50
2.3.3 MULTISTAGE METHODS 53
2.3.4 THREE-LEVEL SCHEMES 56
2.3.5 CONTROLLING THE LEAPFROG COMPUTATIONAL MODE 60
2.3.6 HIGHER-ORDER SCHEMES 65
2.4 SPACE-DIFFERENCING 72
2.4.1 DIFFERENTIAL-DIFFERENCE EQUATIONS AND WAVE DISPERSION 73
2.4.2 DISSIPATION, DISPERSION, AND THE MODIFIED EQUATION . .. 80 2.4.3
ARTIFICIAL DISSIPATION 82
2.4.4 COMPACT DIFFERENCING 86
2.5 COMBINED TIME- AND SPACE-DIFFERENCING 89
2.5.1 THE DISCRETE-DISPERSION RELATION 91
2.5.2 THE MODIFIED EQUATION 94
2.5.3 THE LAX-WENDROFF METHOD 95
2.6 SUMMARY DISCUSSION OF ELEMENTARY METHODS 99
PROBLEMS 101
3 BEYOND THE ONE-WAY WAVE EQUATION 107
3.1 SYSTEMS OF EQUATIONS 107
3.1.1 STABILITY 108
3.1.2 STAGGERED MESHES 113
3.2 THREE OR MORE INDEPENDENT VARIABLES 117
3.2.1 SCALAR ADVECTION IN TWO DIMENSIONS 117
3.2.2 SYSTEMS OF EQUATIONS IN SEVERAL DIMENSIONS 126
3.3 SPLITTING INTO FRACTIONAL STEPS 129
3.3.1 SPLIT EXPLICIT SCHEMES 130
3.3.2 SPLIT IMPLICIT SCHEMES 132
3.3.3 STABILITY OF SPLIT SCHEMES 134
3.4 DIFFUSION, SOURCES, AND SINKS 136
3.4.1 PURE DIFFUSION 136
3.4.2 ADVECTION AND DIFFUSION 138
3.4.3 ADVECTION WITH SOURCES AND SINKS 144
3.5 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 147
3.5.1 ALIASING ERROR 148
3.5.2 CONSERVATION AND STABILITY 154
3.6 NONLINEAR INSTABILITY 159
3.6.1 BURGERS S EQUATION 159
3.6.2 THE BAROTROPIC VORTICITY EQUATION 163
PROBLEMS 167
IMAGE 4
CONTENTS XV
4 SERIES-EXPANSION METHODS 173
4.1 STRATEGIES FOR MINIMIZING THE RESIDUAL 173
4.2 THE SPECTRAL METHOD 176
4.2.1 COMPARISON WITH FINITE-DIFFERENCE METHODS 177
4.2.2 IMPROVING EFFICIENCY USING THE TRANSFORM METHOD . .. 184 4.2.3
CONSERVATION AND THE GALERKIN APPROXIMATION 189 4.3 THE PSEUDOSPECTRAL
METHOD 191
4.4 SPHERICAL HARMONICS 195
4.4.1 TRUNCATING THE EXPANSION 197
4.4.2 ELIMINATION OF THE POLE PROBLEM 200
4.4.3 GAUSSIAN QUADRATURE AND THE TRANSFORM METHOD 202 4.4.4 NONLINEAR
SHALLOW-WATER EQUATIONS 207
4.5 THE FINITE-ELEMENT METHOD 212
4.5.1 GALERKIN APPROXIMATION WITH CHAPEAU FUNCTIONS . . .. 214 4.5.2
PETROV-GALERKIN AND TAYLOR-GALERKIN METHODS 216 4.5.3 QUADRATIC
EXPANSION FUNCTIONS 219
4.5.4 HERMITE-CUBIC EXPANSION FUNCTIONS 226
4.5.5 TWO-DIMENSIONAL EXPANSION FUNCTIONS 231
PROBLEMS 234
5 FINITE VOLUME METHODS 241
5.1 CONSERVATION LAWS AND WEAK SOLUTIONS 243
5.1.1 THE RIEMANN PROBLEM 244
5.1.2 ENTROPY-CONSISTENT SOLUTIONS 246
5.2 FINITE-VOLUME METHODS AND CONVERGENCE 249
5.2.1 MONOTONE SCHEMES 251
5.2.2 TVD METHODS 252
5.3 DISCONTINUITIES IN GEOPHYSICAL FLUID DYNAMICS 254
5.4 FLUX-CORRECTED TRANSPORT 257
5.4.1 FLUX CORRECTION: THE ORIGINAL PROPOSAL 259
5.4.2 THE ZALESAK CORRECTOR 260
5.4.3 ITERATIVE FLUX CORRECTION 263
5.5 FIUX-LIMITER METHODS 263
5.5.1 ENSURING THAT THE SCHEME IS TVD 264
5.5.2 POSSIBLE FLUX LIMITERS 267
5.5.3 FLOW VELOCITIES OF ARBITRARY SIGN 271
5.6 APPROXIMATION WITH LOCAL POLYNOMIALS 272
5.6.1 GODUNOV S METHOD 272
5.6.2 PIECEWISE-LINEAR FUNCTIONS 274
5.7 TWO SPATIAL DIMENSIONS 277
5.7.1 FCT IN TWO DIMENSIONS 277
5.7.2 FLUX-LIMITER METHODS FOR UNIFORM 2-D FLOW 279
5.7.3 NONUNIFORM NONDIVERGENT FLOW 282
5.7.4 A NUMERICAL EXAMPLE 284
5.7.5 WHEN IS A FLUX LIMITER NECESSARY? 291
IMAGE 5
XV I CONTENTS
5.8 SCHEMES FOR POSITIVE DEFINITE ADVECTION 292
5.8.1 AN FCT APPROACH 293
5.8.2 ANTIDIFFUSION VIA UPSTREAM DIFFERENCING 294
5.9 CURVILINEAR COORDINATES 296
PROBLEMS 2 97
6 SEMI-LAGRANGIAN METHODS 303
6.1 THE SCALAR ADVECTION EQUATION 305
6.1.1 CONSTANT VELOCITY 305
6.1.2 VARIABLE VELOCITY 310
6.2 FORCING IN THE LAGRANGIAN FRAME 313
6.3 SYSTEMS OF EQUATIONS 318
6.3.1 COMPARISON WITH THE METHOD OF CHARACTERISTICS 318 6.3.2
SEMI-IMPLICIT SEMI-LAGRANGIAN SCHEMES 320
6.4 ALTERNATIVE TRAJECTORIES 324
6.4.1 A NONINTERPOLATING LEAPFROG SCHEME 325
6.4.2 INTERPOLATION VIA PARAMETRIZED ADVECTION 327
6.5 EULERIAN OR SEMI-LAGRANGIAN? 330
PROBLEMS 331
7 PHYSICALLY INSIGNIFICANT FAST WAVES 335
7.1 THE PROJECTION METHOD 336
7.1.1 FORWARD-IN-TIME IMPLEMENTATION 337
7.1.2 LEAPFROG IMPLEMENTATION 339
7.1.3 SOLVING THE POISSON EQUATION FOR PRESSURE 340
7.2 THE SEMI-IMPLICIT METHOD 342
7.2.1 LARGE TIME STEPS AND POOR ACCURACY 343
7.2.2 A PROTOTYPE PROBLEM 345
7.2.3 SEMI-IMPLICIT SOLUTION OF THE SHALLOW-WATER EQUATIONS . 347 7.2.4
SEMI-IMPLICIT SOLUTION OF THE EULER EQUATIONS 350 7.2.5 NUMERICAL
IMPLEMENTATION 356
7.3 FRACTIONAL-STEP METHODS 359
7.3.1 COMPLETE OPERATOR SPLITTING 359
7.3.2 PARTIALLY SPLIT OPERATORS 365
7.4 SUMMARY OF SCHEMES FOR NONHYDROSTATIC MODELS 371 7.5 THE HYDROSTATIC
APPROXIMATION 372
7.6 PRIMITIVE EQUATION MODELS 374
7.6.1 PRESSURE AND ER COORDINATES 375
7.6.2 SPECTRAL REPRESENTATION OF THE HORIZONTAL STRUCTURE . . . 379
7.6.3 VERTICAL DIFFERENCING 381
7.6.4 ENERGY CONSERVATION 383
7.6.5 SEMI-IMPLICIT TIME-DIFFERENCING 387
PROBLEMS 389
IMAGE 6
CONTENTS XVII
8 NONREFLECTING BOUNDARY CONDITIONS 395
8.1 ONE-DIMENSIONAL FLOW 397
8.1.1 WELL-POSED INITIAL-BOUNDARY VALUE PROBLEMS 397
8.1.2 THE RADIATION CONDITION 400
8.1.3 TIME-DEPENDENT BOUNDARY DATA 401
8.1.4 REFLECTIONS AT AN ARTIFICIAL BOUNDARY- THE CONTINUOUS CASE 402
8.1.5 REFLECTIONS AT AN ARTIFICIAL BOUNDARY- THE DISCRETIZED CASE 403
8.1.6 STABILITY IN THE PRESENCC OF BOUNDARICS 409
8.2 TWO-DIMENSIONAL SHALLOW-WATER FLOW 412
8.2.1 ONE-WAY WAVE EQUATIONS 414
8.2.2 NUMERICAL IMPLEMENTATION 419
8.3 TWO-DIMCNSIONAL STRATIFICD FLOW 419
8.3.1 LATERAL BOUNDARY CONDITIONS 420
8.3.2 UPPCR BOUNDARY CONDITIONS 424
8.3.3 NUMERICAL IMPLEMENTATION OF THE RADIATION UPPCR BOUNDARY CONDITION
429
8.4 WAVE-ABSORBING LAYERS 431
8.5 SUMMARY 436
PROBLEMS 437
APPENDIX NUMERICAL MISCELLANY 439
A.L FINITE-DIFFERENCE OPERATOR NOTATION 439
A.2 TRIDIAGONAL SOLVERS 440
A.2.1 CODE FOR A TRIDIAGONAL SOLVER 440
A.2.2 CODE FOR A PERIODIC TRIDIAGONAL SOLVER 441
BIBLIOGRAPHY 443
INDEX 457
|
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| dewey-ones | 550 - Earth sciences |
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| dewey-search | 550.1532059 |
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| discipline | Geologie / Paläontologie Physik Mathematik Geographie |
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| id | DE-604.BV012408970 |
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| indexdate | 2024-07-09T18:27:05Z |
| institution | BVB |
| isbn | 0387983767 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008418568 |
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| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Durran, Dale R. Verfasser (DE-588)142615110 aut Numerical methods for wave equations in geophysical fluid dynamics Dale R. Durran New York [u.a.] Springer 1999 XVII, 465 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 32 Literaturverz. S. 443 - 455 Geophysik - Hydrodynamik - Wellengleichung - Numerisches Verfahren Geophysik (DE-588)4020252-5 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Hydrodynamik (DE-588)4026302-2 gnd rswk-swf Wellengleichung (DE-588)4065315-8 gnd rswk-swf Geophysik (DE-588)4020252-5 s Hydrodynamik (DE-588)4026302-2 s Wellengleichung (DE-588)4065315-8 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Strömungsmechanik (DE-588)4077970-1 s Texts in applied mathematics 32 (DE-604)BV002476038 32 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008418568&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Durran, Dale R. Numerical methods for wave equations in geophysical fluid dynamics Texts in applied mathematics Geophysik - Hydrodynamik - Wellengleichung - Numerisches Verfahren Geophysik (DE-588)4020252-5 gnd Strömungsmechanik (DE-588)4077970-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Hydrodynamik (DE-588)4026302-2 gnd Wellengleichung (DE-588)4065315-8 gnd |
| subject_GND | (DE-588)4020252-5 (DE-588)4077970-1 (DE-588)4128130-5 (DE-588)4026302-2 (DE-588)4065315-8 |
| title | Numerical methods for wave equations in geophysical fluid dynamics |
| title_auth | Numerical methods for wave equations in geophysical fluid dynamics |
| title_exact_search | Numerical methods for wave equations in geophysical fluid dynamics |
| title_full | Numerical methods for wave equations in geophysical fluid dynamics Dale R. Durran |
| title_fullStr | Numerical methods for wave equations in geophysical fluid dynamics Dale R. Durran |
| title_full_unstemmed | Numerical methods for wave equations in geophysical fluid dynamics Dale R. Durran |
| title_short | Numerical methods for wave equations in geophysical fluid dynamics |
| title_sort | numerical methods for wave equations in geophysical fluid dynamics |
| topic | Geophysik - Hydrodynamik - Wellengleichung - Numerisches Verfahren Geophysik (DE-588)4020252-5 gnd Strömungsmechanik (DE-588)4077970-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Hydrodynamik (DE-588)4026302-2 gnd Wellengleichung (DE-588)4065315-8 gnd |
| topic_facet | Geophysik - Hydrodynamik - Wellengleichung - Numerisches Verfahren Geophysik Strömungsmechanik Numerisches Verfahren Hydrodynamik Wellengleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008418568&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT durrandaler numericalmethodsforwaveequationsingeophysicalfluiddynamics |