346 Pages 10 Color & 79 B/W Illustrations
    by CRC Press

    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
    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
    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
    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
    Poisson Equation
    Poisson Solutions: I
    Poisson Solutions: II
    Test of the Potential
    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
    References appear in each chapter.


    Peter Bodenheimer, Gregory P. Laughlin, Michal Rozyczka, Tomasz Plewa, Harold W. Yorke

    … 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