Nodal discontinuous Galerkin methods: algorithms, analysis, and applications
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2008
|
| Series: | Texts in applied mathematics
54 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 500 Seiten graphische Darstellungen |
| ISBN: | 9780387720654 9780387720678 0387720650 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
1 Introduction 1
1.1 A brief account of history 10
1.2 Summary of the chapters 13
1.3 On the use and abuse of the Matlab codes 16
1.4 Scope of text and audience 16
2 The key ideas 19
2.1 Briefly on notation 19
2.2 Basic elements of the schemes 20
2.2.1 The first schemes 20
2.2.2 An alternative viewpoint 29
2.3 Toward more general formulations 31
2.4 Interlude on linear hyperbolic problems 34
2.5 Exercises 39
3 Making it work in one dimension 43
3.1 Legendre polynomials and nodal elements 43
3.2 Elementwise operations 51
3.3 Getting the grid together and computing the metric 56
3.4 Dealing with time 63
3.5 Putting it all together 64
3.6 Maxwell s equations 67
3.7 Exercises 72
4 Insight through theory 75
4.1 A bit more notation 75
4.2 Briefly on convergence 76
4.3 Approximations by orthogonal polynomials and consistency ... 77
4.4 Stability 83
4.5 Error estimates and error boundedness 85
4.6 Dispersive properties 88
XII Contents
4.7 Discrete stability and timestep choices 93
4.8 Taming the CFL condition 97
4.8.1 Improvements by mapping techniques 98
4.8.2 Filtering by co-volume grids 102
4.8.3 Local timestepping 108
4.9 Exercises 112
5 Nonlinear problems 115
5.1 Conservation laws 115
5.2 The basic schemes and their properties 118
5.3 Aliasing, instabilities, and filter stabilization 123
5.4 Problems on nonconservative form 134
5.5 Error estimates for nonlinear problems with smooth solutions . 135
5.6 Problems with discontinuous solutions 136
5.6.1 Filtering 139
5.6.2 Limiting 145
5.7 Strong stability-preserving Runge-Kutta methods 157
5.8 A few general results 160
5.9 The Euler equations of compressible gas dynamics 161
5.10 Exercises 165
6 Beyond one dimension 169
6.1 Modes and nodes in two dimensions 171
6.2 Elementwise operations 183
6.3 Assembling the grid 190
6.4 Timestepping and boundary conditions 197
6.5 Maxwell s equations 200
6.6 Compressible gas dynamics 206
6.6.1 Variational crimes, aliasing, filtering, and cubature
integration 210
6.6.2 Numerical fluxes revisited 217
6.6.3 Limiters in two dimensions 224
6.7 A few theoretical results 236
6.8 Exercises 239
7 Higher-order equations 243
7.1 Higher-order time-dependent problems 245
7.1.1 The heat equation 245
7.1.2 Extensions to mixed and higher-order problems 255
7.2 Elliptic problems 261
7.2.1 Two-dimensional Poisson and Helmholtz equations .... 275
7.2.2 A look at basic theoretical properties 287
7.3 Intermission of solving linear systems 296
7.3.1 Direct methods 297
7.3.2 Iterative methods 299
Contents XIII
7.4 The incompressible Navier-Stokes equations 300
7.4.1 Temporal splitting scheme 301
7.4.2 The spatial discretization 302
7.4.3 Benchmarks and validations 308
7.5 The compressible Navier-Stokes equations 314
7.5.1 Integration-based gradient, divergence, and jump
operators 316
7.5.2 Solver for the compressible Navier-Stokes equations
of gas dynamics 319
7.5.3 A few test cases 326
7.6 Exercises 327
8 Spectral properties of discontinuous Galerkin operators.... 331
8.1 The Laplace eigenvalue problem 334
8.1.1 Impact of the penalty parameter on the spectrum 338
8.2 The Maxwell eigenvalue problem 342
8.2.1 The two-dimensional eigenvalue problem 348
8.2.2 The three-dimensional eigenproblem 367
8.2.3 Consequences in the time domain 370
9 Curvilinear elements and nonconforming discretizations . .. 373
9.1 Isoparametric curvilinear elements 373
9.1.1 Forming the curvilinear element 375
9.1.2 Building operators on curvilinear elements 378
9.1.3 Maxwell s equations on meshes with curvilinear
elements 383
9.2 Nonconforming discretizations 386
9.2.1 Nonconforming element refinement 387
9.2.2 Nonconforming order refinement 396
9.3 Exercises 405
10 Into the third dimension 407
10.1 Modes and nodes in three dimensions 409
10.2 Elementwise operations 418
10.3 Assembling the grid 425
10.4 Briefly on timestepping 432
10.5 Maxwell s equations 432
10.6 Three-dimensional Poisson equation 437
10.7 Exercises 441
A Appendix A: Jacobi polynomials and beyond 445
A.I Orthonormal polynomials beyond one dimension 448
B Appendix B: Briefly on grid generation 451
B.I Fundamentals 453
B.2 Creating boundary maps 457
XIV Contents
C Appendix C: Software, variables, and helpful scripts 461
C.I List of important variables defined in the codes 461
C.2 List of additional useful scripts 465
References 473
Index 495
|
| adam_txt |
Contents
1 Introduction 1
1.1 A brief account of history 10
1.2 Summary of the chapters 13
1.3 On the use and abuse of the Matlab codes 16
1.4 Scope of text and audience 16
2 The key ideas 19
2.1 Briefly on notation 19
2.2 Basic elements of the schemes 20
2.2.1 The first schemes 20
2.2.2 An alternative viewpoint 29
2.3 Toward more general formulations 31
2.4 Interlude on linear hyperbolic problems 34
2.5 Exercises 39
3 Making it work in one dimension 43
3.1 Legendre polynomials and nodal elements 43
3.2 Elementwise operations 51
3.3 Getting the grid together and computing the metric 56
3.4 Dealing with time 63
3.5 Putting it all together 64
3.6 Maxwell's equations 67
3.7 Exercises 72
4 Insight through theory 75
4.1 A bit more notation 75
4.2 Briefly on convergence 76
4.3 Approximations by orthogonal polynomials and consistency . 77
4.4 Stability " 83
4.5 Error estimates and error boundedness 85
4.6 Dispersive properties 88
XII Contents
4.7 Discrete stability and timestep choices 93
4.8 Taming the CFL condition 97
4.8.1 Improvements by mapping techniques 98
4.8.2 Filtering by co-volume grids 102
4.8.3 Local timestepping 108
4.9 Exercises 112
5 Nonlinear problems 115
5.1 Conservation laws 115
5.2 The basic schemes and their properties 118
5.3 Aliasing, instabilities, and filter stabilization 123
5.4 Problems on nonconservative form 134
5.5 Error estimates for nonlinear problems with smooth solutions . 135
5.6 Problems with discontinuous solutions 136
5.6.1 Filtering 139
5.6.2 Limiting 145
5.7 Strong stability-preserving Runge-Kutta methods 157
5.8 A few general results 160
5.9 The Euler equations of compressible gas dynamics 161
5.10 Exercises 165
6 Beyond one dimension 169
6.1 Modes and nodes in two dimensions 171
6.2 Elementwise operations 183
6.3 Assembling the grid 190
6.4 Timestepping and boundary conditions 197
6.5 Maxwell's equations 200
6.6 Compressible gas dynamics 206
6.6.1 Variational crimes, aliasing, filtering, and cubature
integration 210
6.6.2 Numerical fluxes revisited 217
6.6.3 Limiters in two dimensions 224
6.7 A few theoretical results 236
6.8 Exercises 239
7 Higher-order equations 243
7.1 Higher-order time-dependent problems 245
7.1.1 The heat equation 245
7.1.2 Extensions to mixed and higher-order problems 255
7.2 Elliptic problems 261
7.2.1 Two-dimensional Poisson and Helmholtz equations . 275
7.2.2 A look at basic theoretical properties 287
7.3 Intermission of solving linear systems 296
7.3.1 Direct methods 297
7.3.2 Iterative methods 299
Contents XIII
7.4 The incompressible Navier-Stokes equations 300
7.4.1 Temporal splitting scheme 301
7.4.2 The spatial discretization 302
7.4.3 Benchmarks and validations 308
7.5 The compressible Navier-Stokes equations 314
7.5.1 Integration-based gradient, divergence, and jump
operators 316
7.5.2 Solver for the compressible Navier-Stokes equations
of gas dynamics 319
7.5.3 A few test cases 326
7.6 Exercises 327
8 Spectral properties of discontinuous Galerkin operators. 331
8.1 The Laplace eigenvalue problem 334
8.1.1 Impact of the penalty parameter on the spectrum 338
8.2 The Maxwell eigenvalue problem 342
8.2.1 The two-dimensional eigenvalue problem 348
8.2.2 The three-dimensional eigenproblem 367
8.2.3 Consequences in the time domain 370
9 Curvilinear elements and nonconforming discretizations . . 373
9.1 Isoparametric curvilinear elements 373
9.1.1 Forming the curvilinear element 375
9.1.2 Building operators on curvilinear elements 378
9.1.3 Maxwell's equations on meshes with curvilinear
elements 383
9.2 Nonconforming discretizations 386
9.2.1 Nonconforming element refinement 387
9.2.2 Nonconforming order refinement 396
9.3 Exercises 405
10 Into the third dimension 407
10.1 Modes and nodes in three dimensions 409
10.2 Elementwise operations 418
10.3 Assembling the grid 425
10.4 Briefly on timestepping 432
10.5 Maxwell's equations 432
10.6 Three-dimensional Poisson equation 437
10.7 Exercises 441
A Appendix A: Jacobi polynomials and beyond 445
A.I Orthonormal polynomials beyond one dimension 448
B Appendix B: Briefly on grid generation 451
B.I Fundamentals 453
B.2 Creating boundary maps 457
XIV Contents
C Appendix C: Software, variables, and helpful scripts 461
C.I List of important variables defined in the codes 461
C.2 List of additional useful scripts 465
References 473
Index 495 |
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| index_date | 2024-07-02T20:21:33Z |
| indexdate | 2024-07-09T21:13:44Z |
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| isbn | 9780387720654 9780387720678 0387720650 |
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| series2 | Texts in applied mathematics |
| spelling | Hesthaven, Jan S. Verfasser (DE-588)140381619 aut Nodal discontinuous Galerkin methods algorithms, analysis, and applications Jan S. Hesthaven ; Tim Warburton New York, NY Springer 2008 XIV, 500 Seiten graphische Darstellungen txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 54 Differential equations, Partial Finite element method Galerkin methods Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Galerkin-Methode (DE-588)4155831-5 gnd rswk-swf Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd rswk-swf Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Galerkin-Methode (DE-588)4155831-5 s 1\p DE-604 Warburton, Tim Verfasser aut Texts in applied mathematics 54 (DE-604)BV002476038 54 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016420464&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hesthaven, Jan S. Warburton, Tim Nodal discontinuous Galerkin methods algorithms, analysis, and applications Texts in applied mathematics Differential equations, Partial Finite element method Galerkin methods Partielle Differentialgleichung (DE-588)4044779-0 gnd Galerkin-Methode (DE-588)4155831-5 gnd Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4155831-5 (DE-588)4588309-9 |
| title | Nodal discontinuous Galerkin methods algorithms, analysis, and applications |
| title_auth | Nodal discontinuous Galerkin methods algorithms, analysis, and applications |
| title_exact_search | Nodal discontinuous Galerkin methods algorithms, analysis, and applications |
| title_exact_search_txtP | Nodal discontinuous Galerkin methods algorithms, analysis, and applications |
| title_full | Nodal discontinuous Galerkin methods algorithms, analysis, and applications Jan S. Hesthaven ; Tim Warburton |
| title_fullStr | Nodal discontinuous Galerkin methods algorithms, analysis, and applications Jan S. Hesthaven ; Tim Warburton |
| title_full_unstemmed | Nodal discontinuous Galerkin methods algorithms, analysis, and applications Jan S. Hesthaven ; Tim Warburton |
| title_short | Nodal discontinuous Galerkin methods |
| title_sort | nodal discontinuous galerkin methods algorithms analysis and applications |
| title_sub | algorithms, analysis, and applications |
| topic | Differential equations, Partial Finite element method Galerkin methods Partielle Differentialgleichung (DE-588)4044779-0 gnd Galerkin-Methode (DE-588)4155831-5 gnd Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd |
| topic_facet | Differential equations, Partial Finite element method Galerkin methods Partielle Differentialgleichung Galerkin-Methode Diskontinuierliche Galerkin-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016420464&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT hesthavenjans nodaldiscontinuousgalerkinmethodsalgorithmsanalysisandapplications AT warburtontim nodaldiscontinuousgalerkinmethodsalgorithmsanalysisandapplications |