Analysis and Numerics for Conservation Laws (eBook)

Preface 5
Contents 8
Wave Processes at Interfaces 10
1 Introduction 10
2 Level Set 12
3 Governing Equations and Method of Solution 13
4 Numerical Results 15
5 Conclusion and Future Work 23
References 33
Numerics for Magnetoplasmadynamic Propulsion 35
1 Introduction 35
2 Conservation Equations 36
3 Numerical Methods 38
4 Results 43
5 Summary 45
References 52
Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers 54
1 Introduction 54
2 Structure of fluid flows 56
3 Hexagonal kinetic models 60
4 Kinetic boundary layers 68
References 72
High-resolution Simulation of Detonations with Detailed Chemistry 75
1 Introduction 75
2 Detonation Modeling 77
3 Numerical Methods 81
4 Adaptive Mesh Refinement 86
5 Numerical Results 89
6 Conclusions 94
References 96
Numerical Linear Stability Analysis for Compressible Fluids 98
1 Numerical linear stability analysis 99
2 The compressible Rayleigh-B´enard problem 101
3 The compressible Taylor-Couette problem 104
4 Conclusion 109
References 109
Simulation of Solar Radiative Magneto- Convection 111
1 Introduction 111
2 Equations of radiative magnetohydrodynamics 114
3 Numerical methods for radiative transfer 117
4 Simulations results for solar magneto-convection 127
5 Summary & Outlook
References 139
Riemann Problem for the Euler Equation with Non- Convex Equation of State including Phase Transitions 141
1 Introduction 141
2 Retrograde and BZT Fluids 143
3 Physical Model 144
4 The Riemann Problem of Gas Dynamics 147
5 Numerical Results 159
References 166
Radiation Magnetohydrodynamics: Analysis for Model Problems and Efficient 3d- Simulations for the Full System 167
1 Introduction 167
2 Radiation Hydrodynamics: Analysis 173
3 Design of Efficient Approximate Riemann Solvers 179
4 Coping with the Needs of Atmospheric Simulations 189
5 Approximating the Radiation Source Term 193
6 Dynamic Load Balancing for Complex 3d Simulations 197
7 Conclusions 200
References 201
Kinetic Schemes for Selected Initial and Boundary Value Problems 207
1 Introduction 207
2 The Boltzmann-Peierls Equation 209
3 Relativistic Euler Equations 225
4 Conclusion 233
References 234
A Local Level-Set Method under Involvement of Topological Aspects 237
1 Introduction 237
2 Basic Numerical Concept 238
3 Basics of Level-Set Methods 239
4 Numerical Level-Set Approach 241
5 Numerical Treatment of the Jump Conditions 247
6 Results and Discussion 252
7 Computational Examples 254
8 Conclusion 259
References 260
Hyperbolic Systems and Transport Equations in Mathematical Biology 261
1 Introduction 262
2 Models for Chemotaxis 265
3 Coupled Dynamics and Quiescent States 271
4 Moment Closure 276
References 280
Travelling Waves in Systems of Hyperbolic Balance Laws 284
1 Introduction 284
2 Oscillatory profiles of stiff balance laws 285
3 Bifurcation of heteroclinic waves 293
References 302
The Role of the Jacobian in the Adaptive Discontinuous Galerkin Method for the Compressible Euler Equations 304
1 Introduction 304
2 Discontinuous Galerkin discretization of the compressible Euler equations 307
3 The Jacobian of the numerical scheme 309
4 Numerical results 310
5 Concluding Remarks 318
References 319
The Multi-Scale Dust Formation in Substellar Atmospheres 320
1 Introduction 320
2 The model of a dust forming gas flow 321
3 Dust formation on small scales 326
4 Dust formation on large scales 332
5 Conclusions 336
References 339
Meshless Methods for Conservation Laws 341
1 Introduction 341
2 The Finite Volume Particle Method 342
3 The Finite Pointset Method 352
References 363
Simulations of Turbulent Thermonuclear Burning in Type Ia Supernovae 365
1 Introduction 365
2 The level set method 366
3 Implementation 368
4 Turbulent nuclear burning 369
5 Application to the supernova problem 371
6 Predictions for observable quantities 375
7 Supplementary studies 378
8 Conclusion 384
References 385
Hyperbolic GLM Scheme for Elliptic Constraints in Computational Electromagnetics and MHD 387
1 Introduction 387
2 Symmetries in Maxwell theory 388
3 Field theoretical method for divergence correction 391
4 Numerical Experiment I: Maxwell-Vlasov Equations 396
5 Numerical Experiment II: Shallow Water MHD Equations 397
6 Discussion 403
References 405
Flexible Flame Structure Modelling in a Flame Front Tracking Scheme 407
1 Introduction 408
2 Basic Level-set/in-cell-reconstruction Scheme 410
3 Tracking Flames with Unsteady Internal Structure 415
4 Results 421
5 Summary and conclusions 424
6 Outlook 425
References 425
Riemann-Solver Free Schemes 430
1 Introduction 430
2 Decompositions of Hyperbolic Systems 433
3 Exact and Approximate Evolution Operators for Smooth Solutions 434
4 Comparison with the EG Evolution Operator 438
5 Numerical Results and Comparison 442
6 A Property of the HLL Scheme 446
7 Conclusion 449
References 450
Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes 453
1 Introduction 453
2 An Overview of Relaxation Basics 456
3 Relaxation Dynamics 459
4 Scaling Limits 467
5 Convergence of Relaxation Schemes 469
References 473
Multidimensional Adaptive Staggered Grids 479
1 Introduction 479
2 Finite Volume scheme on staggered grids 480
3 Adaptive staggered grids in 2D 482
4 Adaptive staggered grids in 3D 486
References 492
On Hyperbolic Relaxation Problems 494
1 Introduction 494
2 On relaxation approximations 497
3 Initial-boundary value problems 500
4 Stability conditions and relaxation limits 505
5 Shock structure problems 509
6 Entropy and global solutions 511
References 516
Appendix: Color Plates 520