Numerical Methods in Astrophysics: An

Numerical Methods in Astrophysics: An Introduction outlines various fundamental numerical methods that can solve gravitational dynamics, hydrodynamics, and radiation transport equations. This resource indicates which methods are most suitable for particular problems, demonstrates what the accuracy requirements are in numerical simulations, and suggests ways to test for and reduce the inevitable negative effects.After an introduction to the basic equations and derivations, the book focuses on practical applications of the numerical methods. It explores hydrodynamic problems in one dimension, N-body particle dynamics, smoothed particle hydrodynamics, and stellar structure and evolution. The authors also examine advanced techniques in grid-based hydrodynamics, evaluate the methods for calculating the gravitational forces in an astrophysical system, and discuss specific problems in grid-based methods for radiation transfer. The book incorporates brief user instructions and a CD-ROM of the numerical codes, allowing readers to experiment with the codes to suit their own needs.With numerous examples and sample problems that cover a wide range of current research topics, this highly practical guide illustrates how to solve key astrophysics problems, providing a clear introduction for graduate and undergraduate students as well as researchers and professionals.

Basic Equations	p. 1
The Boltzmann Equation	p. 1
Conservation Laws of Hydrodynamics	p. 6
The Validity of the Continuous Medium Approximation	p. 10
Eulerian and Lagrangian Formulation of Hydrodynamics	p. 12
Viscosity and Navier-Stokes Equations	p. 14
Radiation Transfer	p. 19
Absorption, Emission, and Scattering	p. 21
Moments of the Boltzmann Equation for Photons	p. 22
Optically Thick and Optically Thin Limits	p. 25
Flux-Limited Diffusion	p. 26
Energy Equation in the Optically Thick Limit	p. 27
Conducting and Magnetized Media	p. 29
Maxwell's Equations	p. 29
Equations of Magnetohydrodynamics	p. 32
Limits of the MHD Approximation	p. 33
Field Freezing	p. 34
Summary	p. 36
References	p. 36
Numerical Approximations to Partial Differential Equations	p. 39
Numerical Modeling with Finite-Difference Equations	p. 39
Difference Quotient	p. 41
Discrete Representation of Variables, Functions, and Derivatives	p. 43
Stability of Finite-Difference Methods	p. 49
Physical Meaning of Stability Criterion	p. 52
A Useful Implicit Scheme	p. 58
Diffusion, Dispersion, and Grid Resolution Limit	p. 62
Alternative Methods	p. 65
References	p. 70
N-Body Particle Methods	p. 73
Introduction to the N-Body Problem	p. 73
Euler and Runge-Kutta Methods	p. 74
The Description of Orbital Motion in Terms of Orbital Elements	p. 79
The Few-Body Problem: Bulirsch-Stoer Integration	p. 85
Lyapunov Time Estimation	p. 87
Symplectic Integration	p. 90
N-Body Codes for Large N	p. 94
Close Encounters and Regularization	p. 99
Force Calculation: The Tree Method	p. 104
Force Calculation: Fast Fourier Transforms	p. 107
References	p. 113
Smoothed Particle Hydrodynamics	p. 115
Rudimentary SPH	p. 115
Colliding Planets: An SPH Test Problem	p. 118
Necessary Improvements to Rudimentary SPH	p. 120
Initial Conditions	p. 122
Kernels with Compact Support	p. 122
Combining SPH with a Tree Code	p. 123
Variable Smoothing Lengths	p. 124
A Resolution Requirement	p. 126
Introducing an Energy Equation into SPH	p. 127
Heat Transfer in SPH	p. 128
Shocks in SPH	p. 129
Time Integration	p. 133
Summary	p. 134
References	p. 137
Stellar Evolution	p. 139
Equations for Equilibrium of a Star	p. 140
Radiative, Conductive, and Convective Energy Transfer	p. 141
Change in Chemical Composition	p. 143
Boundary Conditions	p. 144
An Implicit Lagrangian Technique: Henyey Method	p. 147
Physics Packages	p. 155
Equation of State	p. 156
Opacity	p. 158
Nuclear Reactions	p. 160
Examples	p. 164
Evolution of the Sun	p. 164
Age Determination for a Star Cluster	p. 166
References	p. 168
Grid-Based Hydrodynamics	p. 169
Flow Discontinuities and How to Handle Them	p. 170
Steepening of Sound Waves	p. 170
Rankine-Hugoniot Conditions	p. 173
Shock Tube and Riemann Problem	p. 175
Artificial Viscosity	p. 178
A Simple Lagrangian Hydrocode	p. 181
Basic Eulerian Techniques	p. 185
Conservation of Physical Quantities	p. 185
Advection	p. 186
Godunov Method for Calculating Fluxes	p. 188
Operator Splitting	p. 189
Accuracy, Convergence, and Efficiency	p. 191
Adaptive Mesh Refinement	p. 193
A Multidimensional Eulerian Hydrocode	p. 197
Source Terms	p. 200
Advection Terms	p. 201
Boundary Conditions	p. 204
Time Step Control	p. 204
2 1/2-Dimensional Simulations	p. 206
Axial Symmetry	p. 206
Radiation Transport	p. 208
Thin Circumstellar Disk	p. 213
Examples	p. 214
Rayleigh-Taylor Instability	p. 214
Supernova Explosion	p. 216
Protostar Collapse and Disk Formation	p. 217
Spiral Waves in a Thin Self-Gravitating Disk	p. 219
References	p. 221
Poisson Equation	p. 223
Poisson Solutions: I	p. 224
Direct Summation	p. 224
Fourier Methods for Solving Equation (7.4)	p. 225
Self-Consistent Field	p. 230
Poisson Solutions: II	p. 235
Boundary Conditions	p. 236
Alternating Direction Implicit Method	p. 239
Successive Overrelaxation	p. 241
Multigrid Method	p. 243
Fourier Techniques	p. 243
Cyclic Reduction	p. 247
Polynomial Expansions in Three Dimensions	p. 248
Test of the Potential	p. 249
References	p. 251
Magnetohydrodynamics	p. 253
Basic Assumptions and Definitions	p. 253
MHD Source Terms	p. 256
Solving the Induction Equation	p. 259
Initial and Boundary Conditions	p. 264
Examples and Exercises	p. 265
Contraction of a Magnetized Ring	p. 265
Propagation of a Jet with a Helical Field	p. 265
Magnetic Buoyancy Instability	p. 267
Magnetorotational Instability	p. 270
Concluding Remarks	p. 274
References	p. 274
Radiation Transport	p. 277
Solving the Ray Equation for the Continuum	p. 277
Solution for Frequency-Dependent Radiation Transfer in Spherical Symmetry	p. 279
Frequency-Dependent Stellar Atmospheres	p. 285
Technique for Flux-Limited Diffusion in Two Space Dimensions	p. 290
Example: Spectrum of a Rotating, Collapsing Object	p. 300
Example: 3-D Calculations of the Solar Photosphere	p. 305
References	p. 308
Numerical Codes	p. 309
Radiation Transfer	p. 309
Stellar Evolution	p. 311
One-Dimensional Lagrangian Hydro	p. 316
ZEUS: 3-D Hydrodynamics	p. 317
N-Body Codes	p. 318
Smoothed Particle Hydrodynamics	p. 322
References	p. 323
Index	p. 325
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Numerical Methods in Astrophysics: An Introduction

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