1st Edition

Computational Methods in Plasma Physics

By Stephen Jardin Copyright 2010
    372 Pages 61 B/W Illustrations
    by CRC Press

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    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.

    Introduction to Magnetohydrodynamic Equations
    Magnetohydrodynamic (MHD) Equations

    Introduction to Finite Difference Equations
    Implicit and Explicit Methods
    Consistency, Convergence, and Stability
    Von Neumann Stability Analysis
    Accuracy and Conservative Differencing

    Finite Difference Methods for Elliptic Equations
    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
    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
    Magnetic Field, Current, and Surface Functions
    Constructing Flux Coordinates from Ψ(R, Z)
    Inverse Equilibrium Equation

    Diffusion and Transport in Axisymmetric Geometry
    Basic Equations and Orderings
    Equilibrium Constraint
    Time Scales

    Numerical Methods for Parabolic Equations
    One Dimensional Diffusion Equations
    Multiple Dimensions

    Methods of Ideal MHD Stability Analysis
    Basic Equations
    Variational Forms
    Cylindrical Geometry
    Toroidal Geometry

    Numerical Methods for Hyperbolic Equations
    Explicit Centered-Space Methods
    Explicit Upwind Differencing
    Limiter Methods
    Implicit Methods

    Spectral Methods for Initial Value Problems
    Orthogonal Expansion Functions
    Non-Linear Problems
    Time Discretization
    Implicit Example: Gyrofluid Magnetic Reconnection

    The Finite Element Method
    Ritz Method in One Dimension
    Galerkin Method in One Dimension
    Finite Elements in Two Dimensions
    Eigenvalue Problems



    A Summary appears at the end of each chapter.


    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