Computational Methods in Plasma Physics

Computational Methods in Plasma Physics

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Features

  • Presents a unique combination of mathematical techniques and associated computational algorithms needed to perform meaningful simulations of magnetized plasma
  • Gives a comprehensive treatment of the plasma equilibrium problem as well as a unique derivation of methods for solving the transport timescale evolution of magnetized plasma
  • Offers an accessible introduction to many advanced computational methods currently used
  • Covers finite difference, spectral, and finite element methods
  • Contains an extensive overview of the various approaches to solving sparse matrix equations, along with their relative merits and limitations

Summary

Assuming no prior knowledge of plasma physics or numerical methods, Computational Methods in Plasma Physics covers the computational mathematics and techniques needed to simulate magnetically confined plasmas in modern magnetic fusion experiments and future magnetic fusion reactors. Largely self-contained, the text presents the basic concepts necessary for the numerical solution of partial differential equations.

Along with discussing numerical stability and accuracy, the author explores many of the algorithms used today in enough depth so that readers can analyze their stability, efficiency, and scaling properties. He focuses on mathematical models where the plasma is treated as a conducting fluid, since this is the most mature plasma model and most applicable to experiments. The book also emphasizes toroidal confinement geometries, particularly the tokamak—a very successful configuration for confining a high-temperature plasma. Many of the basic numerical techniques presented are also appropriate for equations encountered in a higher-dimensional phase space.

One of the most challenging research areas in modern science is to develop suitable algorithms that lead to stable and accurate solutions that can span relevant time and space scales. This book provides an excellent working knowledge of the algorithms used by the plasma physics community, helping readers on their way to more advanced study.

Table of Contents

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.

Author Bio(s)

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.

Editorial Reviews

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

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