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Tables of Contents for Computational Methods for Astrohphysical Fluid Flow
Chapter/Section Title
Page #
Page Count
Nonlinear Conservation Laws and Finite Volume Methods
1
160
Randall J. LeVeque
1. Introduction
1
7
1.1 Software
3
1
1.2 Notation
4
1
1.3 Classification of Differential Equations
5
3
2. Derivation of Conservation Laws
8
14
2.1 The Euler Equations of Gas Dynamics
10
1
2.2 Dissipative Fluxes
11
1
2.3 Source Terms
11
1
2.4 Radiative Transfer and Isothermal Equations
12
2
2.5 Multi-dimensional Conservation Laws
14
1
2.6 The Shock Tube Problem
15
7
3. Mathematical Theory of Hyperbolic Systems
22
21
3.1 Scalar Equations
22
5
3.2 Linear Hyperbolic Systems
27
5
3.3 Nonlinear Systems
32
8
3.4 The Riemann Problem for the Euler Equations
40
3
4. Numerical Methods in One Dimension
43
41
4.1 Finite Difference Theory
43
9
4.2 Finite Volume Methods
52
3
4.3 Importance of Conservation Form -- Incorrect Shock Speeds
55
1
4.4 Numerical Flux Functions
56
1
4.5 Godunov's Method
56
4
4.6 Approximate Riemann Solvers
60
4
4.7 High-Resolution Methods
64
14
4.8 Other Approaches
78
4
4.9 Boundary Conditions
82
2
5. Source Terms and Fractional Steps
84
17
5.1 Unsplit Methods
85
1
5.2 Fractional Step Methods
86
1
5.3 General Formulation of Fractional Step Methods
87
3
5.4 Stiff Source Terms
90
6
5.5 Quasi-stationary Flow and Gravity
96
5
6. Multi-dimensional Problems
101
10
6.1 Dimensional Splitting
103
1
6.2 Multi-dimensional Finite Volume Methods
103
1
6.3 Grids and Adaptive Refinement
104
7
7. Computational Difficulties
111
7
7.1 Low-Density Flows
111
1
7.2 Discrete Shocks and Viscous Profiles
112
1
7.3 Start-Up Errors
113
2
7.4 Wall Heating
115
1
7.5 Slow-Moving Shocks
115
1
7.6 Grid Orientation Effects
116
1
7.7 Grid-Aligned Shocks
116
2
8. Magnetohydrodynamics
118
14
8.1 The MHD Equations
119
2
8.2 One-Dimensional MHD
121
4
8.3 Solving the Riemann Problem
125
1
8.4 Nonstrict Hyperbolicity
125
2
8.5 Stiffness
127
1
8.6 The Divergence of B
128
2
8.7 Riemann Problems in Multi-dimensional MHD
130
1
8.8 Staggered Grids
131
1
8.9 The 8-Wave Riemann Solver
132
1
9. Relativistic Hydrodynamics
132
16
9.1 Conservation Laws in Spacetime
133
2
9.2 The Continuity Equation
135
1
9.3 The 4-Momentum of a Particle
136
1
9.4 The Stress-Energy Tensor
137
2
9.5 Finite Volume Methods
139
2
9.6 Multi-dimensional Relativistic Flow
141
1
9.7 Gravitation and General Relativity
142
6
References
148
13
Radiation Hydrodynamics
161
102
Dimitri Mihalas
1. Basic Radiation Theory
161
8
1.1 Specific Intensity
161
1
1.2 Photon Number Density
161
1
1.3 Photon Distribution Function
162
1
1.4 Mean Intensity
162
1
1.5 Radiation Energy Density
162
1
1.6 Radiation Energy Flux
163
1
1.7 Radiation Momentum Density
163
1
1.8 Radiation Stress Tensor (Radiation Pressure Tensor)
164
2
1.9 Thermal Radiation
166
2
1.10 Thermodynamics of Thermal Radiation and a Perfect Gas
168
1
2. The Transfer Equation
169
9
2.1 Absorption, Emission, and Scattering
169
2
2.2 The Equation of Transfer
171
3
2.3 Moments of the Transfer Equation
174
4
3. Lorentz Transformation of the Transfer Equation
178
10
3.1 Lorentz Transformation of the Photon 4-Momentum
178
2
3.2 Lorentz Transformation of the Specific Intensity, Opacity, and Emissivity
180
2
3.3 Lorentz Transformation of the Radiation Stress Energy Tensor
182
2
3.4 The Radiation 4-Force Density Vector
184
1
3.5 Covariant Form of the Transfer Equation
185
3
4. Inertial-Frame Equations of Radiation Hydrodynamics
188
11
4.1 Inertial-Frame Radiation Equations
188
6
4.2 Inertial-Frame Equations of Radiation Hydrodynamics
194
5
5. Comoving-Frame Equation of Transfer
199
12
5.1 Special Relativistic Derivation (D. Mihalas)
199
6
5.2 Consistency Between Comoving-Frame and Inertial-Frame Equations
205
1
5.3 Noninertial Frame Derivation (J.I. Castor)
206
4
5.4 Analysis of O(v/c) Terms
210
1
6. Lagrangian Equations of Radiation Hydrodynamics
211
8
6.1 Momentum Equation
211
1
6.2 Gas Energy Equation
212
1
6.3 First Law of Thermodynamics for the Radiation Field
213
1
6.4 First Law of Thermodynamics for the Radiating Fluid
213
1
6.5 Mechanical Energy Equation
214
1
6.6 Total Energy Equation
214
2
6.7 Consistency of Different Forms of the Radiating-Fluid Energy and Momentum Equations
216
1
6.8 Consistency of Inertial-Frame and Comoving-Frame Radiation Energy and Momentum Equations
217
2
7. Radiation Diffusion
219
15
7.1 Radiation Diffusion
219
7
7.2 Nonequilibrium Diffusion
226
5
7.3 The Problem of Flux Limiting
231
3
8. Shock Propagation: Numerical Methods
234
11
8.1 Acoustic Waves
234
1
8.2 Numerical Stability
235
1
8.3 Systems of Equations
236
2
8.4 Implications of Shock Development
238
1
8.5 Implications of Diffusive Energy Transport
239
2
8.6 Illustrative Example
241
4
9. Numerical Radiation Hydrodynamics
245
9
9.1 Radiating Fluid Energy and Momentum Equations
245
2
9.2 Computational Strategy
247
2
9.3 Energy Conservation
249
1
9.4 Formal Solution
249
2
9.5 Multigroup Equations
251
1
9.6 An Astrophysical Example
251
3
10. Adaptive-Grid Radiation Hydrodynamics
254
6
10.1 Front Fitting
254
1
10.2 Artificial Dissipation
255
1
10.3 The Adaptive Grid
255
4
10.4 The TITAN Code
259
1
References
260
3
Radiation Hydrodynamics: Numerical Aspects and Applications
263
80
Ernst A. Dorfi
1. Introduction
263
4
1.1 General Remarks on the Numerical Method
263
1
1.2 Time Scales
264
1
1.3 Length Scales
264
1
1.4 Interaction Between Matter and Radiation
265
1
1.5 Moving Fronts
266
1
2. Basic Equations
267
8
2.1 Radiation Hydrodynamics (RHD)
267
2
2.2 Coupling Terms
269
1
2.3 Closure Conditions
269
2
2.4 Opacity
271
1
2.5 Equation of State
272
2
2.6 Transport Theorem
274
1
3. Solution Strategy
275
16
3.1 Integral Form of the RHD Equations
275
2
3.2 Symbolic Notation
277
1
3.3 Moving Coordinates
277
1
3.4 Implicit Discretization
277
2
3.5 Time-centering
279
1
3.6 Adaptive RHD Equations
280
1
3.7 Discretization of Gradients and Divergence Terms
280
1
3.8 Diffusion
281
1
3.9 Advection
282
1
3.10 Initial Conditions
283
1
3.11 Boundary Conditions
284
1
3.12 Artificial Viscosity
285
1
3.13 Discrete RHD Equations
286
2
3.14 Radiative Closure Condition
288
2
3.15 Radiative Boundary Conditions
290
1
3.16 Eddington Factor
290
1
4. Adaptive Grids
291
16
4.1 Basic Grid Properties
292
1
4.2 Desired Resolution
292
1
4.3 Spatial and Temporal Smoothing
293
1
4.4 Grid Equation
294
1
4.5 Grid Boundary Conditions
295
1
4.6 Grid Motion
296
1
4.7 Remarks on the Grid Equation
296
1
4.8 First Example: Simple Test Function
297
1
4.9 Second Example: Shock Tube Problem
298
5
4.10 Initial Grid Distributions
303
4
5. Further Computational Needs
307
5
5.1 Rational Spline Interpolation
307
1
5.2 CPU-Time Requirements
308
1
5.3 Iteration Procedure and Matrix Inversion
309
1
5.4 Structure of the Jacobi Matrix
310
2
5.5 Time-Step Control
312
1
6. Computational Examples
312
22
6.1 Evolution of Supernova Remnants (SNRs)
312
7
6.2 Nonlinear Stellar Pulsations
319
7
6.3 Protostellar Collapse
326
3
6.4 Dust-Driven Winds
329
3
6.5 Radiative Transfer
332
2
7. Discussion
334
6
7.1 Internal Accuracy
334
1
7.2 Problems
335
1
7.3 Advantages and Disadvantages of the implicit formulation
336
1
7.4 Nuclear and Chemical Networks and Convection
337
1
7.5 Multidimensional Versions
338
1
7.6 Improvements and Further Recommendations
338
2
References
340
3
Simulation of Astrophysical Fluid Flow
343
152
Ewald Muller
1. Introduction
343
1
2. Simulations: A Link Between Observation and Theory
344
9
2.1 Procedure and Resources
346
2
2.2 Some Basic Issues
348
5
3. Simulations of Core Collapse Supernovae
353
52
3.1 Supernova Observations
353
7
3.2 Physics of Spherical Core Collapse
360
11
3.3 Observations Demanding Nonspherical Models
371
3
3.4 Rayleigh-Taylor Instabilities in Supernova Envelopes
374
1
3.5 Simulations of RT Instabilities in Supernova Envelopes
375
2
3.6 Neutrino Driven Convective Instabilities
377
5
3.7 Rotational Core Collapse
382
10
3.8 Gravitational Wave Signature of Core Collapse Supernovae
392
13
4. Hydrodynamics and Thermonuclear Burning
405
32
4.1 Time Scales
406
2
4.2 Types of Burning
408
11
4.3 Nuclear Reaction Networks
419
6
4.4 Coupling Reaction Networks and Hydrodynamics
425
6
4.5 Some Instructive Numerical Experiments
431
6
5. Simulation of Astrophysical Jets
437
26
5.1 Observations of Extragalactic Jets
439
4
5.2 Newtonian Hydrodynamic Simulations of Extragalactic Jets
443
2
5.3 Morphology and Dynamics
445
6
5.4 Relativistic Simulations
451
5
5.5 Morphology and Dynamics of Relativistic Jets
456
4
5.6 Long Term Evolution of Relativistic Jets
460
2
5.7 Simulation of Parsec-Scale Jets
462
1
6. Smoothed Particle Hydrodynamics
463
17
6.1 The SPH formalism
464
3
6.2 Self-gravity
467
3
6.3 Variable Smoothing Length
470
2
6.4 Time Integration, Initial Model
472
1
6.5 Computational Aspects
473
1
6.6 How dissipative is SPH?
474
2
6.7 How Well Does SPH Treat Shocks?
476
4
References
480
15
Index
495