Computational Methods for Astrophysical Fluid Flow: Saas-Fee Advanced Course 27. Lecture Notes 1997 Swiss Society for Astrophysics and Astronomy

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Springer Science & Business Media, Aug 19, 1998 - Science - 510 pages
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This book leads directly to the most modern numerical techniques for compressible fluid flow, with special consideration given to astrophysical applications. Emphasis is put on high-resolution shock-capturing finite-volume schemes based on Riemann solvers. The applications of such schemes, in particular the PPM method, are given and include large-scale simulations of supernova explosions by core collapse and thermonuclear burning and astrophysical jets. Parts two and three treat radiation hydrodynamics. The power of adaptive (moving) grids is demonstrated with a number of stellar-physical simulations showing very crispy shock-front structures.
 

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Contents

Nonlinear Conservation Laws and Finite Volume Methods
1
11 Software
3
12 Notation
4
13 Classification of Differential Equations
5
2 Derivation of Conservation Laws
8
21 The Euler Equations of Gas Dynamics
10
22 Dissipative Fluxes
11
24 Radiative Transfer and Isothermal Equations
12
92 Computational Strategy
247
93 Energy Conservation
249
95 Multigroup Equations
251
10 AdaptiveGrid Radiation Hydrodynamics
254
102 Artificial Dissipation
255
104 The TITAN Code
259
References
260
Numerical Aspects and Applications
263

25 Multidimensional Conservation Laws
14
26 The Shock Tube Problem
15
3 Mathematical Theory of Hyperbolic Systems
22
32 Linear Hyperbolic Systems
27
33 Nonlinear Systems
32
34 The Riemann Problem for the Euler Equations
40
4 Numerical Methods in One Dimension
43
42 Finite Volume Methods
52
43 Importance of Conservation Form Incorrect Shock Speeds
55
44 Numerical Flux Functions
56
46 Approximate Riemann Solvers
60
47 HighResolution Methods
64
48 Other Approaches
78
49 Boundary Conditions
82
5 Source Terms and Fractional Steps
84
51 Unsplit Methods
85
52 Fractional Step Methods
86
53 General Formulation of Fractional Step Methods
87
54 Stiff Source Terms
90
55 Quasistationary Flow and Gravity
96
6 Multidimensional Problems
101
61 Dimensional Splitting
103
63 Grids and Adaptive Refinement
104
7 Computational Difficulties
111
72 Discrete Shocks and Viscous Profiles
112
73 StartUp Errors
113
74 Wall Heating
115
76 Grid Orientation Effects
116
8 Magnetohydrodynamics
118
81 The MHD Equations
119
82 OneDimensional MHD
121
83 Solving the Riemann Problem
125
85 Stiffness
127
86 The Divergence of B
128
87 Riemann Problems in Multidimensional MHD
130
88 Staggered Grids
131
89 The 8Wave Riemann Solver
132
91 Conservation Laws in Spacetime
133
92 The Continuity Equation
135
93 The 4Momentum of a Particle
136
94 The StressEnergy Tensor
137
95 Finite Volume Methods
139
96 Multidimensional Relativistic Flow
141
97 Gravitation and General Relativity
142
References
148
Radiation Hydrodynamics
161
13 Photon Distribution Function
162
16 Radiation Energy Flux
163
18 Radiation Stress Tensor Radiation Pressure Tensor
164
19 Thermal Radiation
166
110 Thermodynamics of Thermal Radiation and a Perfect Gas
168
2 The Transfer Equation
169
22 The Equation of Transfer
171
23 Moments of the Transfer Equation
174
3 Lorentz Transformation of the Transfer Equation
178
32 Lorentz Transformation of the Specific Intensity Opacity and Emissivity
180
33 Lorentz Transformation of the Radiation Stress Energy Tensor
182
34 The Radiation 4Force Density Vector
184
35 Covariant Form of the Transfer Equation
185
4 InertialFrame Equations of Radiation Hydrodynamics
188
42 InertialFrame Equations of Radiation Hydrodynamics
194
5 ComovingFrame Equation of Transfer
199
52 Consistency Between ComovingFrame and InertialFrame Equations
205
53 Noninertial Frame Derivation JI Castor
206
54 Analysis of Ovc Terms
210
6 Lagrangian Equations of Radiation Hydrodynamics
211
62 Gas Energy Equation
212
63 First Law of Thermodynamics for the Radiation Field
213
65 Mechanical Energy Equation
214
67 Consistency of Different Forms of the RadiatingFluid Energy and Momentum Equations
216
68 Consistency of InertialFrame and ComovingFrame Radiation Energy and Momentum Equations
217
7 Radiation Diffusion
219
72 Nonequilibrium Diffusion
226
73 The Problem of Flux Limiting
231
Numerical Methods
234
82 Numerical Stability
235
83 Systems of Equations
236
84 Implications of Shock Development
238
85 Implications of Diffusive Energy Transport
239
86 Illustrative Example
241
9 Numerical Radiation Hydrodynamics
245
12 Time Scales
264
14 Interaction Between Matter and Radiation
265
15 Moving Fronts
266
2 Basic Equations
267
22 Coupling Terms
269
24 Opacity
271
25 Equation of State
272
26 Transport Theorem
274
3 Solution Strategy
275
32 Symbolic Notation
277
35 Timecentering
279
36 Adaptive RHD Equations
280
38 Diffusion
281
39 Advection
282
310 Initial Conditions
283
311 Boundary Conditions
284
312 Artificial Viscosity
285
313 Discrete RHD Equations
286
314 Radiative Closure Condition
288
315 Radiative Boundary Conditions
290
4 Adaptive Grids
291
41 Basic Grid Properties
292
43 Spatial and Temporal Smoothing
293
44 Grid Equation
294
45 Grid Boundary Conditions
295
46 Grid Motion
296
Simple Test Function
297
Shock Tube Problem
298
410 Initial Grid Distributions
303
5 Further Computational Needs
307
52 CPUTime Requirements
308
53 Iteration Procedure and Matrix Inversion
309
54 Structure of the Jacobi Matrix
310
55 TimeStep Control
312
62 Nonlinear Stellar Pulsations
319
63 Protostellar Collapse
326
64 DustDriven Winds
329
65 Radiative Transfer
332
7 Discussion
334
72 Problems
335
73 Advantages and Disadvantages of the implicit formulation
336
74 Nuclear and Chemical Networks and Convection
337
75 Multidimensional Versions
338
References
340
Simulation of Astrophysical Fluid Flow
343
A Link Between Observation and Theory
344
21 Procedure and Resources
346
22 Some Basic Issues
348
3 Simulations of Core Collapse Supernovae
353
32 Physics of Spherical Core Collapse
360
33 Observations Demanding Nonspherical Models
371
34 RayleighTaylor Instabilities in Supernova Envelopes
374
35 Simulations of RT Instabilities in Supernova Envelopes
375
36 Neutrino Driven Convective Instabilities
377
37 Rotational Core Collapse
382
38 Gravitational Wave Signature of Core Collapse Supernovae
392
4 Hydrodynamics and Thermonuclear Burning
405
41 Time Scales
406
42 Types of Burning
408
43 Nuclear Reaction Networks
419
44 Coupling Reaction Networks and Hydrodynamics
425
45 Some Instructive Numerical Experiments
431
5 Simulation of Astrophysical Jets
437
51 Observations of Extragalactic Jets
439
52 Newtonian Hydrodynamic Simulations of Extragalactic Jets
443
53 Morphology and Dynamics
445
54 Relativistic Simulations
451
55 Morphology and Dynamics of Relativistic Jets
456
56 Long Term Evolution of Relativistic Jets
460
57 Simulation of ParsecScale Jets
462
6 Smoothed Particle Hydrodynamics
463
61 The SPH formalism
464
62 Selfgravity
467
63 Variable Smoothing Length
470
64 Time Integration Initial Model
472
65 Computational Aspects
473
66 How dissipative is SPH?
474
67 How Well Does SPH Treat Shocks?
476
References
480
Index
495
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