Hyperbolic partial differential equations: theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ]
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Braunschweig [u.a.]
Vieweg
2002
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| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XI, 320 S. graph. Darst. |
| ISBN: | 3528031883 |
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| 245 | 1 | 0 | |a Hyperbolic partial differential equations |b theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] |c Andreas Meister ...[eds.] [Summerschool on Hyperbolic Partial Differential Equations] |
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Record in the Search Index
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| adam_text | vii
Contents
1 Hyperbolic Conservation Laws and Industrial Applications 1
1.1 Transport theorem and balance laws 1
1.1.1 The flux vector 1
1.1.2 The transport theorem 3
1.1.3 Continuum hypothesis 5
1.1.4 Examples of balance laws 6
Population model 6
Traffic model 6
Kinetic equations 8
Euler equation 10
1.2 Linear initial and boundary value problems 12
1.2.1 The method of characteristics 12
Advection equation 14
Stationary radiative transport 16
Vlasov equation with constant force 19
Crystal precipitation 19
1.2.2 Linear systems 21
Symmetric hyperbolic systems 21
A problem with a moving boundary 23
1.3 Weak solutions and entropy 27
1.3.1 Weak solutions 30
1.3.2 Entropy condition 36
1.3.3 Ion etching in semiconductor production 39
1.4 Systems of conservation laws 44
1.4.1 Characteristic surfaces 44
1.4.2 The Saint Venant system 45
1.4.3 Rarefaction waves 49
1.4.4 Shock waves 51
viii Contents
Bibliography 57
2 Central Schemes and Systems of Balance Laws 59
2.1 Second order central schemes 60
2.1.1 Hyperbolic systems 60
2.1.2 Conservative schemes 62
2.1.3 Godunov scheme 64
2.1.4 The Nessyahu Tadmor scheme 68
2.2 High order central schemes 71
2.2.1 Time evolution: Runge Kutta methods with Natural Continuous
Extension 72
2.2.2 The Central CWENO (CWENO) Reconstruction 74
2.2.3 Numerical results 79
Scalar equation 79
Systems of equations 80
2.2.4 Improvements 83
2.2.5 Semidiscrete central schemes 85
2.3 Multidimensional central schemes 86
2.3.1 Multidimensional CWENO reconstruction 89
2.3.2 The Reconstruction from Cell Averages 89
2.3.3 The Bi Quadratic Polynomials 92
2.3.4 The Weights 92
2.3.5 The Reconstruction of Flux Derivatives 94
2.3.6 The Algorithm 96
2.3.7 Systems of Equations 96
2.3.8 Numerical Examples 97
Two Dimensional Gas Dynamics Equations 98
2.3.9 Improvements 100
2.4 Treatment of the source 102
Further developments 108
Bibliography 110
ix
3 Methods on unstructured grids, WENO and ENO Recovery techniquesll5
3.1 Introduction to finite volume approximations 115
3.2 Governing equations 115
3.3 Finite volume approximations 118
3.3.1 Conforming triangulations 119
3.3.2 Primary grid methods for the Euler equations 121
3.3.3 Box methods for the Euler equations 123
3.3.4 Remarks on the spatial accuracy 125
3.3.5 Numerical examples 127
3.3.6 Discretisation of the viscous fluxes 133
3.4 Time stepping schemes 139
3.4.1 The CFL condition 139
3.4.2 Explicit discretisations 140
3.4.3 Implicit time discretisations 141
3.5 Remarks on the philosophy of ENO schemes 144
3.6 Polynomial recovery 145
3.6.1 Stencil selection algorithms 146
3.6.2 Computation of the recovery polynomial 148
3.6.3 Recovery on primary grids 152
3.6.4 Recovery for box methods 160
3.7 WENO approximations 165
3.8 The theory of optimal recovery 166
3.8.1 Basic notions 168
3.8.2 Splines 173
3.9 Grid adaptivity for box methods 179
3.9.1 A refinement procedure 179
3.9.2 Red Green Refinement 180
Algorithm 1 181
3.9.3 History 181
3.9.4 Recoarsening 182
Algorithm 2 183
Algorithm 3 184
3.9.5 Conservativity 186
x Contents
3.10 Error and residual 187
3.11 Experience with I? 189
3.11.1 On the necessity of proper weighting 189
3.11.2 The weighted I,2 norm 195
3.11.3 Unsteady Problems 199
3.12 The dual graph norm 203
3.12.1 Friedrichs systems 203
3.12.2 Symmetrising the Euler equations 204
3.12.3 Cell error and transported error 208
3.12.4 A weak a posteriori error estimate 209
3.12.5 A discrete dual graph norm indicator 213
3.12.6 The error indicator 215
3.12.7 Numerical experiments 219
3.12.8 Problems with the dual graph norm 224
3.13 Closing of the circle: 1? meets dual graph norm 228
Bibliography 229
4 Pressure Correction Methods for all Flow Speeds 233
4.1 Introduction 233
4.2 Conservation Equations 233
4.3 Pressure Correction Equation for Incompressible Flows 236
4.3.1 SIMPLE Method 237
4.3.2 SIMPLEC Method 238
4.3.3 PISO Method 239
4.3.4 SIMPLER Method 240
4.3.5 Under Relaxation Strategies 241
4.3.6 Fractional Step Methods 242
4.4 Pressure Correction Equation for Compressible Flows 243
4.5 Solution Algorithm for all Flow Speeds 245
4.6 FV Method for Arbitrary Control Volumes 246
4.7 Pressure Correction Algorithm for FV Methods 249
4.8 Implementation of Boundary Conditions 252
4.9 Examples of Application 254
xi
4.9.1 Subsonic Laminar Flow Around Airfoil 255
4.9.2 Subsonic Turbulent Flow Around Airfoil 256
4.9.3 Transonic Flow Around Airfoil 259
4.9.4 Supersonic Flow Around Airfoil 259
4.9.5 Flow Through an Orifice 261
4.10 Conclusions 262
Bibliography 267
5 Computational Fluid Dynamics and Aeroacoustics for Low Mach Num¬
ber Flow 269
5.1 Introduction 269
5.2 Non Dimensionalisation of the Governing Equations 272
5.3 The Incompressible Limit of a Compressible Fluid Flow 273
5.4 Numerical Methods for Low Mach Number Fluid Flow 275
5.4.1 Compressible Pressure Correction Schemes 276
5.4.2 Multiple Pressure Variables (MPV )Approach 279
5.4.3 Preconditioning 283
5.5 Sound Generation and Sound Propagation 284
5.5.1 The Equations of Linear Acoustics 284
5.5.2 Lighthill s Acoustic Analogy 285
5.5.3 The Janzen Rayleigh Expansion 286
5.5.4 Expansion around Incompressible Flow 287
5.5.5 Numerical Example: Co Rotating Vortex Pair 289
5.6 Multiple Scale Considerations 290
5.6.1 Outer Asymptotic Expansion 291
5.6.2 Multiple Scale Asymptotic Expansion 292
5.6.3 MPV Approach for Thermo Acoustic Applications 295
5.6.4 Multiple Scale Considerations of Noise Generation 298
5.7 Numerical Aeroacoustics 301
5.8 Conclusions 314
Bibliography 317
|
| adam_txt |
vii
Contents
1 Hyperbolic Conservation Laws and Industrial Applications 1
1.1 Transport theorem and balance laws 1
1.1.1 The flux vector 1
1.1.2 The transport theorem 3
1.1.3 Continuum hypothesis 5
1.1.4 Examples of balance laws 6
Population model 6
Traffic model 6
Kinetic equations 8
Euler equation 10
1.2 Linear initial and boundary value problems 12
1.2.1 The method of characteristics 12
Advection equation 14
Stationary radiative transport 16
Vlasov equation with constant force 19
Crystal precipitation 19
1.2.2 Linear systems 21
Symmetric hyperbolic systems 21
A problem with a moving boundary 23
1.3 Weak solutions and entropy 27
1.3.1 Weak solutions 30
1.3.2 Entropy condition 36
1.3.3 Ion etching in semiconductor production 39
1.4 Systems of conservation laws 44
1.4.1 Characteristic surfaces 44
1.4.2 The Saint Venant system 45
1.4.3 Rarefaction waves 49
1.4.4 Shock waves 51
viii Contents
Bibliography 57
2 Central Schemes and Systems of Balance Laws 59
2.1 Second order central schemes 60
2.1.1 Hyperbolic systems 60
2.1.2 Conservative schemes 62
2.1.3 Godunov scheme 64
2.1.4 The Nessyahu Tadmor scheme 68
2.2 High order central schemes 71
2.2.1 Time evolution: Runge Kutta methods with Natural Continuous
Extension 72
2.2.2 The Central CWENO (CWENO) Reconstruction 74
2.2.3 Numerical results 79
Scalar equation 79
Systems of equations 80
2.2.4 Improvements 83
2.2.5 Semidiscrete central schemes 85
2.3 Multidimensional central schemes 86
2.3.1 Multidimensional CWENO reconstruction 89
2.3.2 The Reconstruction from Cell Averages 89
2.3.3 The Bi Quadratic Polynomials 92
2.3.4 The Weights 92
2.3.5 The Reconstruction of Flux Derivatives 94
2.3.6 The Algorithm 96
2.3.7 Systems of Equations 96
2.3.8 Numerical Examples 97
Two Dimensional Gas Dynamics Equations 98
2.3.9 Improvements 100
2.4 Treatment of the source 102
Further developments 108
Bibliography 110
ix
3 Methods on unstructured grids, WENO and ENO Recovery techniquesll5
3.1 Introduction to finite volume approximations 115
3.2 Governing equations 115
3.3 Finite volume approximations 118
3.3.1 Conforming triangulations 119
3.3.2 Primary grid methods for the Euler equations 121
3.3.3 Box methods for the Euler equations 123
3.3.4 Remarks on the spatial accuracy 125
3.3.5 Numerical examples 127
3.3.6 Discretisation of the viscous fluxes 133
3.4 Time stepping schemes 139
3.4.1 The CFL condition 139
3.4.2 Explicit discretisations 140
3.4.3 Implicit time discretisations 141
3.5 Remarks on the philosophy of ENO schemes 144
3.6 Polynomial recovery 145
3.6.1 Stencil selection algorithms 146
3.6.2 Computation of the recovery polynomial 148
3.6.3 Recovery on primary grids 152
3.6.4 Recovery for box methods 160
3.7 WENO approximations 165
3.8 The theory of optimal recovery 166
3.8.1 Basic notions 168
3.8.2 Splines 173
3.9 Grid adaptivity for box methods 179
3.9.1 A refinement procedure 179
3.9.2 Red Green Refinement 180
Algorithm 1 181
3.9.3 History 181
3.9.4 Recoarsening 182
Algorithm 2 183
Algorithm 3 184
3.9.5 Conservativity 186
x Contents
3.10 Error and residual 187
3.11 Experience with I? 189
3.11.1 On the necessity of proper weighting 189
3.11.2 The weighted I,2 norm 195
3.11.3 Unsteady Problems 199
3.12 The dual graph norm 203
3.12.1 Friedrichs systems 203
3.12.2 Symmetrising the Euler equations 204
3.12.3 Cell error and transported error 208
3.12.4 A weak a posteriori error estimate 209
3.12.5 A discrete dual graph norm indicator 213
3.12.6 The error indicator 215
3.12.7 Numerical experiments 219
3.12.8 Problems with the dual graph norm 224
3.13 Closing of the circle: 1? meets dual graph norm 228
Bibliography 229
4 Pressure Correction Methods for all Flow Speeds 233
4.1 Introduction 233
4.2 Conservation Equations 233
4.3 Pressure Correction Equation for Incompressible Flows 236
4.3.1 SIMPLE Method 237
4.3.2 SIMPLEC Method 238
4.3.3 PISO Method 239
4.3.4 SIMPLER Method 240
4.3.5 Under Relaxation Strategies 241
4.3.6 Fractional Step Methods 242
4.4 Pressure Correction Equation for Compressible Flows 243
4.5 Solution Algorithm for all Flow Speeds 245
4.6 FV Method for Arbitrary Control Volumes 246
4.7 Pressure Correction Algorithm for FV Methods 249
4.8 Implementation of Boundary Conditions 252
4.9 Examples of Application 254
xi
4.9.1 Subsonic Laminar Flow Around Airfoil 255
4.9.2 Subsonic Turbulent Flow Around Airfoil 256
4.9.3 Transonic Flow Around Airfoil 259
4.9.4 Supersonic Flow Around Airfoil 259
4.9.5 Flow Through an Orifice 261
4.10 Conclusions 262
Bibliography 267
5 Computational Fluid Dynamics and Aeroacoustics for Low Mach Num¬
ber Flow 269
5.1 Introduction 269
5.2 Non Dimensionalisation of the Governing Equations 272
5.3 The Incompressible Limit of a Compressible Fluid Flow 273
5.4 Numerical Methods for Low Mach Number Fluid Flow 275
5.4.1 Compressible Pressure Correction Schemes 276
5.4.2 Multiple Pressure Variables (MPV )Approach 279
5.4.3 Preconditioning 283
5.5 Sound Generation and Sound Propagation 284
5.5.1 The Equations of Linear Acoustics 284
5.5.2 Lighthill's Acoustic Analogy 285
5.5.3 The Janzen Rayleigh Expansion 286
5.5.4 Expansion around Incompressible Flow 287
5.5.5 Numerical Example: Co Rotating Vortex Pair 289
5.6 Multiple Scale Considerations 290
5.6.1 Outer Asymptotic Expansion 291
5.6.2 Multiple Scale Asymptotic Expansion 292
5.6.3 MPV Approach for Thermo Acoustic Applications 295
5.6.4 Multiple Scale Considerations of Noise Generation 298
5.7 Numerical Aeroacoustics 301
5.8 Conclusions 314
Bibliography 317 |
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| spelling | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] Andreas Meister ...[eds.] [Summerschool on Hyperbolic Partial Differential Equations] 1. ed. Braunschweig [u.a.] Vieweg 2002 XI, 320 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturangaben Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content (DE-588)4123623-3 Lehrbuch gnd-content Hyperbolische Differentialgleichung (DE-588)4131213-2 s DE-604 Meister, Andreas 1966- Sonstige (DE-588)135617839 oth Summerschool on Hyperbolic Partial Differential Equations (2001, Hamburg-Harburg) Sonstige (DE-588)16146586-9 oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015179919&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd |
| subject_GND | (DE-588)4131213-2 (DE-588)1071861417 (DE-588)4123623-3 |
| title | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] |
| title_auth | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] |
| title_exact_search | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] |
| title_exact_search_txtP | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] |
| title_full | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] Andreas Meister ...[eds.] [Summerschool on Hyperbolic Partial Differential Equations] |
| title_fullStr | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] Andreas Meister ...[eds.] [Summerschool on Hyperbolic Partial Differential Equations] |
| title_full_unstemmed | Hyperbolic partial differential equations theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] Andreas Meister ...[eds.] [Summerschool on Hyperbolic Partial Differential Equations] |
| title_short | Hyperbolic partial differential equations |
| title_sort | hyperbolic partial differential equations theory numerics and applications march 2001 at the technical university of hamburg harburg in germany |
| title_sub | theory, numerics and applications ; [ March 2001 ] ; [ at the Technical University of Hamburg-Harburg in Germany ] |
| topic | Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd |
| topic_facet | Hyperbolische Differentialgleichung Konferenzschrift Lehrbuch |
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