Analysis and Numerics for Conservation Laws (eBook)

Gerald Warnecke (Herausgeber)

eBook Download: PDF

2005 | 2005
X, 542 Seiten
Springer Berlin (Verlag)
978-3-540-27907-5 (ISBN)

Infos

Whatdoasupernovaexplosioninouterspace,?owaroundanairfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite di?erent. So are the motivations for studying them scienti?cally. The super- 8 nova is a thermo-nuclear explosion on a scale of 10 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In ?ows around airfoils of commercial airliners at the scale of 3 10 cm shock waves occur that in?uence the stability of the wings as well as fuel consumption in ?ight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical 1 process, and must be avoided since it damages motors. The scale is 10 cm and these processes must be optimized for e?ciency and environmental conside- tions. The common thread is that the underlying ?uid ?ows may at a certain scale of observation be described by basically the same type of hyperbolic s- tems of partial di?erential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scienti?c progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial di?erential equationsareamajor?eldofresearchinmathematics. Asubstantialportionof mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more spacedimensionsstillposeoneofthemainchallengestomodernmathematics.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 5.12.2005
• Zusatzinfo: X, 542 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Astronomie / Astrophysik
• Themenwelt: Technik ► Bauwesen
• Schlagworte: astrophysics • Calculus • computational astrophysics • computational fluid dynamics • convection • differential equation • Dynamical Systems • Dynamics • Fluid Dynamics • hyperbolic conservation laws • Magnetohydrodynamics • Numerical Methods • Numerics • partial differential equation • stability • Waves
• ISBN-10: 3-540-27907-5 / 3540279075
• ISBN-13: 978-3-540-27907-5 / 9783540279075