Introduction to Magnetohydrodynamic Equations
Introduction
Magnetohydrodynamic (MHD) Equations
Characteristics
Introduction to Finite Difference Equations
Introduction
Implicit and Explicit Methods
Errors
Consistency, Convergence, and Stability
Von Neumann Stability Analysis
Accuracy and Conservative Differencing
Finite Difference Methods for Elliptic Equations
Introduction
One Dimensional Poisson’s Equation
Two Dimensional Poisson’s Equation
Matrix Iterative Approach
Physical Approach to Deriving Iterative Methods
Multigrid Methods
Krylov Space Methods
Finite Fourier Transform
Plasma Equilibrium
Introduction
Derivation of the Grad–Shafranov Equation
The Meaning of Ψ
Exact Solutions
Variational Forms of the Equilibrium Equation
Free Boundary Grad–Shafranov Equation
Experimental Equilibrium Reconstruction
Magnetic Flux Coordinates in a Torus
Introduction
Preliminaries
Magnetic Field, Current, and Surface Functions
Constructing Flux Coordinates from Ψ(R, Z)
Inverse Equilibrium Equation
Diffusion and Transport in Axisymmetric Geometry
Introduction
Basic Equations and Orderings
Equilibrium Constraint
Time Scales
Numerical Methods for Parabolic Equations
Introduction
One Dimensional Diffusion Equations
Multiple Dimensions
Methods of Ideal MHD Stability Analysis
Introduction
Basic Equations
Variational Forms
Cylindrical Geometry
Toroidal Geometry
Numerical Methods for Hyperbolic Equations
Introduction
Explicit Centered-Space Methods
Explicit Upwind Differencing
Limiter Methods
Implicit Methods
Spectral Methods for Initial Value Problems
Introduction
Orthogonal Expansion Functions
Non-Linear Problems
Time Discretization
Implicit Example: Gyrofluid Magnetic Reconnection
The Finite Element Method
Introduction
Ritz Method in One Dimension
Galerkin Method in One Dimension
Finite Elements in Two Dimensions
Eigenvalue Problems
Bibliography
Index
A Summary appears at the end of each chapter.
Biography
Stephen Jardin is a Principal Research Physicist at the Princeton Plasma Physics Laboratory, where he is head of the Theoretical Magnetohydrodynamics Division and co-head of the Computational Plasma Physics Group. He is also a professor in the Department of Astrophysical Sciences at Princeton University and Director and Principal Investigator of the SciDAC Center for Extended Magnetohydrodynamic Modeling. Dr. Jardin is the primary developer of several widely used fusion plasma simulation codes and is currently a U.S. member of the International Tokamak Physics Activity that advises the physics staff of ITER, the world’s largest fusion experiment.
This book provides a comprehensive and self-contained introduction to the computational methods used in plasma physics. The author successfully familiarizes readers with the basic concepts of numerical methods for partial differential equations and conjoins these methods with the magnetohydrodynamic equations that are used in plasma physics. … The extensive treatment of the material, the problems in each chapter, and the accurate topic presentation in this book make it an appropriate textbook for graduate students in physics and engineering with no prior knowledge of plasma physics or numerical mathematics. … great textbook on a highly complex scientific subject. I highly recommend this book …
—Computing Reviews, January 2011
