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Numerical Methods in Astrophysics : An Introduction - Peter Bodenheimer

Hardcover Published: 13th December 2006
ISBN: 9780750308830
Number Of Pages: 344

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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.

... a very thorough introduction ... the book is ideal for a postgraduate student just beginning a Ph.D. in numerical astrophysics or for an undergraduate with a numerical project. However, it also offers more advanced researchers and professionals [with] a clear and useful reminder of the important issues involved in numerical algorithms. ... The codes make an interesting addition to the book in that they allow the reader to actually try out ... some of the numerical algorithms discussed in the book. ...
-Matthew Bate, Geophysical and Astrophysical Fluid Dynamics

The sweep of the book is impressive given its size. Even with the space constraint, room has been found for excellent discussions of code stability, starting with very simple examples, and including nice comparative discussions for various techniques . . . This is a most welcome and carefully thought out book that should help in the search for deeper subterranean seams.
-James Collett, Physical Sciences Educational Review, 2007, Vol. 8, No. 1

Basic Equationsp. 1
The Boltzmann Equationp. 1
Conservation Laws of Hydrodynamicsp. 6
The Validity of the Continuous Medium Approximationp. 10
Eulerian and Lagrangian Formulation of Hydrodynamicsp. 12
Viscosity and Navier-Stokes Equationsp. 14
Radiation Transferp. 19
Absorption, Emission, and Scatteringp. 21
Moments of the Boltzmann Equation for Photonsp. 22
Optically Thick and Optically Thin Limitsp. 25
Flux-Limited Diffusionp. 26
Energy Equation in the Optically Thick Limitp. 27
Conducting and Magnetized Mediap. 29
Maxwell's Equationsp. 29
Equations of Magnetohydrodynamicsp. 32
Limits of the MHD Approximationp. 33
Field Freezingp. 34
Summaryp. 36
Referencesp. 36
Numerical Approximations to Partial Differential Equationsp. 39
Numerical Modeling with Finite-Difference Equationsp. 39
Difference Quotientp. 41
Discrete Representation of Variables, Functions, and Derivativesp. 43
Stability of Finite-Difference Methodsp. 49
Physical Meaning of Stability Criterionp. 52
A Useful Implicit Schemep. 58
Diffusion, Dispersion, and Grid Resolution Limitp. 62
Alternative Methodsp. 65
Referencesp. 70
N-Body Particle Methodsp. 73
Introduction to the N-Body Problemp. 73
Euler and Runge-Kutta Methodsp. 74
The Description of Orbital Motion in Terms of Orbital Elementsp. 79
The Few-Body Problem: Bulirsch-Stoer Integrationp. 85
Lyapunov Time Estimationp. 87
Symplectic Integrationp. 90
N-Body Codes for Large Np. 94
Close Encounters and Regularizationp. 99
Force Calculation: The Tree Methodp. 104
Force Calculation: Fast Fourier Transformsp. 107
Referencesp. 113
Smoothed Particle Hydrodynamicsp. 115
Rudimentary SPHp. 115
Colliding Planets: An SPH Test Problemp. 118
Necessary Improvements to Rudimentary SPHp. 120
Initial Conditionsp. 122
Kernels with Compact Supportp. 122
Combining SPH with a Tree Codep. 123
Variable Smoothing Lengthsp. 124
A Resolution Requirementp. 126
Introducing an Energy Equation into SPHp. 127
Heat Transfer in SPHp. 128
Shocks in SPHp. 129
Time Integrationp. 133
Summaryp. 134
Referencesp. 137
Stellar Evolutionp. 139
Equations for Equilibrium of a Starp. 140
Radiative, Conductive, and Convective Energy Transferp. 141
Change in Chemical Compositionp. 143
Boundary Conditionsp. 144
An Implicit Lagrangian Technique: Henyey Methodp. 147
Physics Packagesp. 155
Equation of Statep. 156
Opacityp. 158
Nuclear Reactionsp. 160
Examplesp. 164
Evolution of the Sunp. 164
Age Determination for a Star Clusterp. 166
Referencesp. 168
Grid-Based Hydrodynamicsp. 169
Flow Discontinuities and How to Handle Themp. 170
Steepening of Sound Wavesp. 170
Rankine-Hugoniot Conditionsp. 173
Shock Tube and Riemann Problemp. 175
Artificial Viscosityp. 178
A Simple Lagrangian Hydrocodep. 181
Basic Eulerian Techniquesp. 185
Conservation of Physical Quantitiesp. 185
Advectionp. 186
Godunov Method for Calculating Fluxesp. 188
Operator Splittingp. 189
Accuracy, Convergence, and Efficiencyp. 191
Adaptive Mesh Refinementp. 193
A Multidimensional Eulerian Hydrocodep. 197
Source Termsp. 200
Advection Termsp. 201
Boundary Conditionsp. 204
Time Step Controlp. 204
2 1/2-Dimensional Simulationsp. 206
Axial Symmetryp. 206
Radiation Transportp. 208
Thin Circumstellar Diskp. 213
Examplesp. 214
Rayleigh-Taylor Instabilityp. 214
Supernova Explosionp. 216
Protostar Collapse and Disk Formationp. 217
Spiral Waves in a Thin Self-Gravitating Diskp. 219
Referencesp. 221
Poisson Equationp. 223
Poisson Solutions: Ip. 224
Direct Summationp. 224
Fourier Methods for Solving Equation (7.4)p. 225
Self-Consistent Fieldp. 230
Poisson Solutions: IIp. 235
Boundary Conditionsp. 236
Alternating Direction Implicit Methodp. 239
Successive Overrelaxationp. 241
Multigrid Methodp. 243
Fourier Techniquesp. 243
Cyclic Reductionp. 247
Polynomial Expansions in Three Dimensionsp. 248
Test of the Potentialp. 249
Referencesp. 251
Magnetohydrodynamicsp. 253
Basic Assumptions and Definitionsp. 253
MHD Source Termsp. 256
Solving the Induction Equationp. 259
Initial and Boundary Conditionsp. 264
Examples and Exercisesp. 265
Contraction of a Magnetized Ringp. 265
Propagation of a Jet with a Helical Fieldp. 265
Magnetic Buoyancy Instabilityp. 267
Magnetorotational Instabilityp. 270
Concluding Remarksp. 274
Referencesp. 274
Radiation Transportp. 277
Solving the Ray Equation for the Continuump. 277
Solution for Frequency-Dependent Radiation Transfer in Spherical Symmetryp. 279
Frequency-Dependent Stellar Atmospheresp. 285
Technique for Flux-Limited Diffusion in Two Space Dimensionsp. 290
Example: Spectrum of a Rotating, Collapsing Objectp. 300
Example: 3-D Calculations of the Solar Photospherep. 305
Referencesp. 308
Numerical Codesp. 309
Radiation Transferp. 309
Stellar Evolutionp. 311
One-Dimensional Lagrangian Hydrop. 316
ZEUS: 3-D Hydrodynamicsp. 317
N-Body Codesp. 318
Smoothed Particle Hydrodynamicsp. 322
Referencesp. 323
Indexp. 325
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780750308830
ISBN-10: 0750308834
Series: Series in Astronomy and Astrophysics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 344
Published: 13th December 2006
Publisher: Taylor & Francis Ltd
Country of Publication: GB
Dimensions (cm): 23.5 x 15.9  x 2.54
Weight (kg): 0.64
Edition Number: 1

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