Numerical Methods in Astrophysics
An Introduction
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Book Description
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.
Table of Contents
Basic Equations
The Boltzmann Equation
Conservation Laws of Hydrodynamics
The Validity of the Continuous Medium Approximation
Eulerian and Lagrangian Formulation of Hydrodynamics
Viscosity and Navier–Stokes Equations
Radiation Transfer
Conducting and Magnetized Media
Numerical Approximations to Partial Differential Equations
Numerical Modeling with Finite-Difference Equations
Difference Quotient
Discrete Representation of Variables, Functions, and Derivatives
Stability of Finite-Difference Methods
Physical Meaning of Stability Criterion
A Useful Implicit Scheme
Diffusion, Dispersion, and Grid Resolution Limit
Alternative Methods
N-Body Particle Methods
Introduction to the N-Body Problem
Euler and Runge–Kutta Methods
The Description of Orbital Motion in Terms of Orbital Elements
The Few-Body Problem: Bulirsch–Stoer Integration
Lyapunov Time Estimation
Symplectic Integration
N-Body Codes for Large N
Close Encounters and Regularization
Force Calculation: The Tree Method
Force Calculation: Fast Fourier Transforms
Smoothed Particle Hydrodynamics
Rudimentary SPH
Colliding Planets: An SPH Test Problem
Necessary Improvements to Rudimentary SPH
Summary
Stellar Evolution
Equations for Equilibrium of a Star
Radiative, Conductive, and Convective Energy Transfer
Change in Chemical Composition
Boundary Conditions
An Implicit Lagrangian Technique: Henyey Method
Physics Packages
Examples
Grid-Based Hydrodynamics
Flow Discontinuities and How to Handle Them
A Simple Lagrangian Hydrocode
Basic Eulerian Techniques
Adaptive Mesh Refinement
A Multidimensional Eulerian Hydrocode
2 1/2-Dimensional Simulations
Examples
Poisson Equation
Poisson Solutions: I
Poisson Solutions: II
Test of the Potential
Magnetohydrodynamics
Basic Assumptions and Definitions
MHD Source Terms
Solving the Induction Equation
Initial and Boundary Conditions
Examples and Exercises
Concluding Remarks
Radiation Transport
Solving the Ray Equation for the Continuum
Solution for Frequency-Dependent Radiation Transfer in Spherical Symmetry
Frequency-Dependent Stellar Atmospheres
Technique for Flux-Limited Diffusion in Two Space Dimensions
Example: Spectrum of a Rotating, Collapsing Object
Example: 3-D Calculations of the Solar Photosphere
Numerical Codes
Radiation Transfer
Stellar Evolution
One-Dimensional Lagrangian Hydro
ZEUS: 3-D Hydrodynamics
N-Body Codes
Smoothed Particle Hydrodynamics
INDEX
References appear in each chapter.
Reviews
… 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 DynamicsThe 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