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About this book
About this book
Numerical methods have become a major research tool in astrophysics, and astrophysics students need to be fully conversant with a variety of mathematical and computational techniques. This highly practical guide discusses some of the major problems in astrophysics that have to be solved numerically, requiring hydrodynamics, MHD and radiative transfer for their solution. It summarizes the advantages and disadvantages of various techniques that could be used on a given problem, and includes examples from recently published solutions and basic codes, which can be developed into research tools. The book includes a full range of sample problems for the student, making it an ideal course text.
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.