NUMERICAL METHODS IN ASTROPHYSICS An Introduction

Transcription

1 -1 Series in Astronomy and Astrophysics NUMERICAL METHODS IN ASTROPHYSICS An Introduction Peter Bodenheimer University of California Santa Cruz, USA Gregory P. Laughlin University of California Santa Cruz, USA Michai Rözyczka Nicolaus Copernicus Astronomical Center Warsaw, Poland HaroldW.Yorke Jet Propulsion Laboratory Pasadena, California, USA Taylor & Francis Taylor & Francis Group New York London

2 Contents Chapter 1 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 Absorption, Emission, and Scattering Moments of the Boltzmann Equation for Photons Optically Thick and Optically Thin Limits Flux-Limited Diffusion Energy Equation in the Optically Thick Limit Conducting and Magnetized Media Maxwell's Equations Equations of Magnetohydrodynamics Limits of the MHD Approximation Field Freezing Summary 36 References 36 Chapter 2 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 65 References 70 Chapter 3 /V-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 99

3 3.9 Force Calculation: The Tree Method Force Calculation: Fast Fourier Transforms j 07 References j j^ Chapter 4 Smoothed Particle Hydrodynamics j Rudimentary SPH Colliding Planets: An SPH Test Problem Necessary Improvements to Rudimentary SPH Initial Conditions Kernels with Compact Support Combining SPH with a Tree Code Variable Smoothing Lengths A Resolution Requirement Introducing an Energy Equation into SPH Heat Transfer in SPH Shocks in SPH Time Integration Summary 134 References 137 Chapter 5 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 Equation of State Opacity Nuclear Reactions Examples Evolution of the Sun Age Determination for a Star Cluster 166 References 168 Chapter 6 Grid-Based Hydrodynamics Flow Discontinuities and How to Handle Them Steepening of Sound Waves Rankine-Hugoniot Conditions Shock Tube and Riemann Problem Artificial Viscosity A Simple Lagrangian Hydrocode Basic Eulerian Techniques Conservation of Physical Quantities 185

4 6.3.2 Advection ] Godunov Method for Calculating Fluxes Operator Splitting Accuracy, Convergence, and Efficiency Adaptive Mesh Reflnement A Multidimensional Eulerian Hydrocode Source Terms Advection Terms Boundary Conditions Time Step Control i-Dimensional Simulations Axial Symmetry Radiation Transport Thin Circumstellar Disk Examples Rayleigh-Taylor Instability 2J Supernova Explosion Protostar Collapse and Disk Formation Spiral Waves in a Thin Self-Gravitating Disk 219 References 221 Chapter 7 Poisson Equation Poisson Solutions: Direct Summation Fourier Methods for Solving Equation (7.4) Self-Consistent Field Poisson Solutions: II Boundary Conditions Alternating Direction Implicit Method Successive Overrelaxation Multigrid Method Fourier Techniques Cyclic Reduction Polynomial Expansions in Three Dimensions Test of the Potential 249 References 251 Chapter 8 Magnetohydrodynamics Basic Assumptions and Definitions MHD Source Terms Solving the Induction Equation Initial and Boundary Conditions Examples and Exercises Contraction of a Magnetized Ring Propagation of a Jet with a Helical Field 265

5 8.5.3 Magnetic Buoyancy Instability Magnetorotational Instability Concluding Remarks 274 References 274 Chapter 9 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 305 References 308 Chapter 10 Numerical Codes Radiation Transfer Stellar Evolution One-Dimensional Lagrangian Hydro ZEUS: 3-D Hydrodynamics N-Body Codes Smoothed Particle Hydrodynamics 322 References 323 Index 325