Table Of ContentSaas-Fee Advanced Course 27
Lecture Notes 1997
Springer
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R. J. LeVeque D. Mihalas
E.A.Dorfi E.MuUer
Computational Methods
for Astrophysical
Fluid Row
Saas-Fee Advanced Course 27
Lecture Notes 1997
Swiss Society for Astrophysics and Astronomy
Edited by O. Steiner and A.Gautschy
With 124 Figures
Springer
Professor R. J. LeVeque Professor E. A. Dorfi
Department of Applied Mathematics Institut fiir Astronomie der Universitat Wien
University of Washington Turkenschanzstrasse 17,
Box 352420, Seattle, WA 98195-2420, USA A-1180 Wien, Austria
rjl @amath.washington.edu, ead @astro.ast.uni vie.ac.at,
http://www.amath.washingion.edu/~rjl/ http://amok.ast.univie.ac.at/'~cad/ead_home.html
Professor D. Mihalas Professor E, Miiller
Department of Astronomy Max-Planck-Institut fiir Astrophysik
University of Illinois Karl-Schwarzschild-Strasse 1
1002 West Green Street D-85740 Garching, Germany
Urbana, IL6I801,USA
Preface
Many standard numerical algorithms for fluid dynamics have their roots
in astrophysics; the reasons are probably twofold. First, since astrophysical
objects are normally not amenable to experimental studies, astrophysicists
seek some understanding by simulating them by methods of computational
physics. With the advancement of computer technology and numerical al-
gorithms, complex astrophysical phenomena such as supernova explosions,
accretion of material onto a star, stellar pulsations, or the granular pattern
of solax convection axe now accessible to simulation almost as if they were ap-
proachable via experiments in the laboratory. Second, the extreme conditions
that prevail in astrophysical flows such as extremely high shock strengths, or
very high compressibility of the fluid lead to the development of novel nu-
merical methods that would normally not have been required for terrestrial
engineering applications.
Computational astrophysics has become an important branch of astro-
physical research and many students and researchers of astrophysics are at
some point in their career confronted with computer simulation results, or
the prospect of executing simulation calculations, or even the writing of a
simulation code. Considering the wealth of numerical schemes and computer
codes available for astrophysical fluid flow, for novices it is not easy to assess
and evaluate simulation results or to choose the correct scheme and to avoid
at least "commonly known" pitfalls when carrying out their own simulations.
The present book should provide some help in such circumstances. More-
over, it might also prove to be a valuable reference for the more experienced
computational astrophysicist.
The first part of this book leads directly to the most modern numerical
techniques for compressible fluid flow, with special consideration given to as-
trophysical applications. Emphasis is put on high-resolution shock-capturing
finite-volume schemes based on Riemann solvers. Examples of unphysical
solutions resulting from lackluster methods and incorrect appUcations are
discussed, as well as more advanced topics such as MHD-Riemann solver
or computational methods for (general) relativistic fluid flow. An extensive
literature list leads the reader to specific topics.
The applications of such schemes, in particular the PPM method, is por-
trayed in the last part of the book (the colorful finale). Examples of large-scale
simulations include supernova explosions by core collapse and thermonuclear
burning and astrophysical jets. They demonstrate the interplay of observa-
VI
tions with simulations and exemplify the deeper physical understanding of
astrophysical objects that can be gained through computer simulations.
For most astrophysical fluid flow, radiation transfer needs to be (should
be) taicen into account, hence part two and three, which treat the daring
subject of radiation hydrodynamics. Dimitri Mihalas presents - in a light-
hearted style - the basic equations of "radiation hydro" that were otherwise
only accessible in his voluminous "Foundations of Radiation Hydrodynamics"
(now out of print). Mihalas' contribution is largely identical to his handwrit-
ten notes that he uses in teaching radiation hydrodynamics at the University
of Illinois at Urbana Champaign. The numerical implementations of these
equations and simulation examples are the subject of part three. The power
of adaptive (moving) grids, which are capable locally refining the resolution
by several orders of magnitude, is demonstrated with a number of stellar-
physical simulations showing very crispy shock-front structures. Interestingly,
these powerful radiation hydrodynamic codes are based on more traditional
finite volume techniques using artificial viscosity and it seems that Riemann
solver methods have not yet been applied in this difficult field of computa-
tional astrophysics.
This book is a written version of the the lectures delivered by Ernst Dorfi,
Randall LeVeque, and Ewald Miiller during the 27th "Saas-Fee Advanced
Course". Dimitri Mihalas, who was scheduled as a lecturer, had to withdraw
shortly before the course began because of an accident which forced him
to remain in hospital for several weeks. The organizers of the course were
very glad when Ernst Dorfi accepted to step in as a lecturer on numerical
radiation hydrodynamics and when Dimitri Mihalas offered to contribute
his (not delivered) lectures to the present book. Ernst Dorfi wrote a more
comprehensive text on the subject of his ad hoc lecture.
The 27th "Saas-Fee Advanced Course" of the Swiss Society for Astro-
physics and Astronomy took place in Les Diablerets, a small village in the
Swiss Alps, during March 3-8, 1997. Ninety participants from 14 countries
attended the course. The Eurotel provided a much appreciated hospitality.
As can be deduced from the photographs in the book, sun and snow were
also abundant and appropriately enjoyed.
This course would not have been possible without the financial contribu-
tions from the Swiss Society for Astrophysics and Astronomy (through the
Swiss Academy of Sciences) which are gratefully acknowledged. O.S. would
like to acknowledge the generous support by the High Altitude Observatory
of the National Center for Atmospheric Research. A.G. was financially sup-
ported by the Swiss National Science Foundation through a PR0FIL2 fellow-
ship. Last but not least we are grateful to Wolfgang Loffler for professionally
maintaining the Saas-Fee Web site.
Freiburg i.Br., Basel Oskar Steiner
May, 1998 Alfred Gautschy
Table of Contents
Nonlinear Conservation Laws
and Finite Volume Methods
Randal J. LeVeque 1
1. Introduction 1
1.1 Software 3
1.2 Notation 4
1.3 Clasification of Diferential Equations 5
2. Derivation of Conservation Laws 8
2.1 The Euler Equations of Gas Dynamics 10
2.2 Disipative Fluxes 1
2.3 Source Terms 1
2.4 Radiative Transfer and Isothermal Equations 12
2.5 Multi-dimensional Conservation Laws 14
2.6 The Shock Tube Problem 15
3. Mathematical Theory of Hyperbolic Systems 2
3.1 Scalar Equations 2
3.2 Linear Hyperbolic Systems 27
3.3 Nonlinear Systems 32
3.4 The Rieman Problem for the Euler Equations 40
4. Numerical Methods in One Dimension 43
4.1 Finite Diference Theory 43
4.2 Finite Volume Methods 52
4.3 Importance of Conservation Form — Incorrect Shock Speeds .. 55
4.4 Numerical Flux Functions 56
4.5 Godunov's Method 56
4.6 Aproximate Rieman Solvers 60
4.7 High-Resolution Methods 64
4.8 Other Aproaches 78
4.9 Boundary Conditions 82
5. Source Terms and Fractional Steps 84
5.1 Unsplit Methods 85
5.2 Fractional Step Methods 86
5.3 General Formulation of Fractional Step Methods 87
5.4 Stif Source Terms 90
VIII Table of Contents
5.5 Quasi-stationary Flow and Gravity 96
6. Multi-dimensional Problems 101
6.1 Dimensional Spliting 103
6.2 Multi-dimensional Finite Volume Methods 103
6.3 Grids and Adaptive Refinement 104
7. Computational Dificulties I l l
7.1 Low-Density Flows I l l
7.2 Discrete Shocks and Viscous Profiles 12
7.3 Start-Up Erors 13
7.4 Wal Heating 15
7.5 Slow-Moving Shocks 15
7.6 Grid Orientation Efects 16
7.7 Grid-Aligned Shocks 16
8. Magnetohydrodynamics 18
8.1 The MHD Equations 19
8.2 One-Dimensional MHD 121
8.3 Solving the Rieman Problem 125
8.4 Nonstrict Hyperbolicity 125
8.5 Stifnes 127
8.6 The Divergence of B 128
8.7 Rieman Problems in Multi-dimensional MHD 130
8.8 Stagered Grids 131
8.9 The 8-Wave Rieman Solver 132
9. Relativistic Hydrodynamics 132
9.1 Conservation Laws in Spacetime 13
9.2 The Continuity Equation 135
9.3 The 4-Momentum of a Particle 136
9.4 The Stres-Energy Tensor 137
9.5 Finite Volume Methods 139
9.6 Multi-dimensional Relativistic Flow 141
9.7 Gravitation and General Relativity 142
References 148
Radiation Hydrodynamics
Dimitri Mihalas 161
1. Basic Radiation Theory 161
1.1 Specific Intensity 161
1.2 Photon Number Density 161
1.3 Photon Distribution Function 162
1.4 Mean Intensity 162
1.5 Radiation Energy Density 162
1.6 Radiation Energy Flux 163
1.7 Radiation Momentum Density 163
1.8 Radiation Stres Tensor (Radiation Presure Tensor) 164
Table of Contents IX
1.9 Thermal Radiation 16
1.10 Thermodynamics of Thermal Radiation and a Perfect Gas . . . . 168
2. The Transfer Equation 169
2.1 Absorption, Emision, and Scatering 169
2.2 The Equation of Transfer 171
2.3 Moments of the Transfer Equation 174
3. Lorentz Transformation of the Transfer Equation 178
3.1 Lorentz Transformation of the Photon 4-Momentum 178
3.2 Lorentz Transformation of the Specific Litensity, Opacity,
and Emisivity 180
3.3 Lorentz Transformation of the Radiation Stress Energy Tensor. 182
3.4 The Radiation 4-Force Density Vector 184
3.5 Covariant Form of the Transfer Equation 185
4. Inertial-Prame Equations of Radiation Hydrodynamics 18
4.1 Inertial-Prame Radiation Equations 18
4.2 Inertial-Prame Equations of Radiation Hydrodynamics 194
5. Comoving-Prame Equation of Transfer 19
5.1 Special Relativistic Derivation (D. Mihalas) 19
5.2 Consistency Between Comoving-Prame
and Inertial-Prame Equations 205
5.3 Noninertial Prame Derivation (J.I. Castor) 206
5.4 Analysis of 0{vlc) Terms 210
6. Lagrangian Equations of Radiation Hydrodynamics 21
6.1 Momentum Equation 21
6.2 Gas Energy Equation 212
6.3 Pirst Law of Thermodynamics for the Radiation Pield 213
6.4 Pirst Law of Thermodynamics for the Radiating Pluid 213
6.5 Mechanical Energy Equation 214
6.6 Total Energy Equation 214
6.7 Consistency of Different Porms of the Radiating-Pluid
Energy and Momentum Equations 216
6.8 Consistency of Inertial-Prame and Comoving-Prame
Radiation Energy and Momentum Equations 217
7. Radiation Difusion 219
7.1 Radiation Difusion 219
7.2 NonequiUbrium Difusion 26
7.3 The Problem of Plux Limiting 231
8. Shock Propagation: Numerical Methods 234
8.1 Acoustic Waves 234
8.2 Numerical Stabihty 235
8.3 Systems of Equations 236
8.4 Implications of Shock Development 238
8.5 Implications of Difusive Energy Transport 239
8.6 Ilustrative Example 241
X Table of Contents
9. Numerical Radiation Hydrodynamics 245
9.1 Radiating Fluid Energy and Momentum Equations 245
9.2 Computational Strategy 247
9.3 Energy Conservation 249
9.4 Formal Solution 249
9.5 Multigroup Equations 251
9.6 An Astrophysical Example 251
10. Adaptive-Grid Radiation Hydrodynamics 254
10.1 Front Fiting 254
10.2 Artificial Disipation 25
10.3 The Adaptive Grid 25
10.4 The TITAN Code 259
References 260
Radiation Hydrodynamics:
Numerical Aspects and Applications
Ernst A. Dorfi 263
1. Introduction 263
1.1 General Remarks on the Numerical Method 263
1.2 Time Scales 264
1.3 Length Scales 264
1.4 Interaction Betwen Mater and Radiation 265
1.5 Moving Fronts 26
2. Basic Equations 267
2.1 Radiation Hydrodynamics (RHD) 267
2.2 Coupling Terms 269
2.3 Closure Condition 269
2.4 Opacity 271
2.5 Equation of State 272
2.6 Transport Theorem 274
3. Solution Strategy 275
3.1 Integral Form of the RHD Equations 275
3.2 Symbolic Notation 27
3.3 Moving Cordinates 27
3.4 Implicit Discretization 27
3.5 Time-centering 279
3.6 Adaptive RHD Equations 280
3.7 Discretization of Gradients and Divergence Terms 280
3.8 Difusion 281
3.9 Advection 282
3.10 Initial Conditions 283
3.1 Boundary Conditions 284
3.12 Artificial Viscosity 285
3.13 Discrete RHD Equations 286
