| Description |
1 online resource (xxii, 349 pages) : illustrations |
| Series |
Chapman & Hall/CRC computational science series |
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Chapman & Hall/CRC computational science series.
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| Contents |
880-01 Introduction to magnetohydrodynamic equations -- Introduction to finite difference equations -- Finite difference methods for elliptic equations -- Plasma equilibrium -- Magnetic flux coordinates in a torus -- Diffusion and transport in axisymmetric geometry -- Numerical methods for parabolic equations -- Methods of ideal MHD stability analysis -- Numerical methods for hyperbolic equations -- Spectral methods for initial value problems -- The finite element method |
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880-01/(S Machine generated contents note: 1. Introduction to Magnetohydrodynamic Equations -- 1.1. Introduction -- 1.2. Magnetohydrodynamic (MHD) Equations -- 1.2.1. Two-Fluid MHD -- 1.2.2. Resistive MHD -- 1.2.3. Ideal MHD -- 1.2.4. Other Equation Sets for MHD -- 1.2.5. Conservation Form -- 1.2.6. Boundary Conditions -- 1.3. Characteristics -- 1.3.1. Characteristics in Ideal MHD -- 1.3.2. Wave Dispersion Relation in Two-Fluid MHD -- 1.4. Summary -- 2. Introduction to Finite Difference Equations -- 2.1. Introduction -- 2.2. Implicit and Explicit Methods -- 2.3. Errors -- 2.4. Consistency, Convergence, and Stability -- 2.5. Von Neumann Stability Analysis -- 2.5.1. Relation to Truncation Error -- 2.5.2. Higher-Order Equations -- 2.5.3. Multiple Space Dimensions -- 2.6. Accuracy and Conservative Differencing -- 2.7. Summary -- 3. Finite Difference Methods for Elliptic Equations -- 3.1. Introduction -- 3.2. One-Dimensional Poisson's Equation -- 3.2.1. Boundary Value Problems in One Dimension -- 3.2.2. Tridiagonal Algorithm -- 3.3. Two-Dimensional Poisson's Equation -- 3.3.1. Neumann Boundary Conditions -- 3.3.2. Gauss Elimination -- 3.3.3. Block-Tridiagonal Method -- 3.3.4. General Direct Solvers for Sparse Matrices -- 3.4. Matrix Iterative Approach -- 3.4.1. Convergence -- 3.4.2. Jacobi's Method -- 3.4.3. Gauss-Seidel Method -- 3.4.4. Successive Over-Relaxation Method (SOR) -- 3.4.5. Convergence Rate of Jacobi's Method -- 3.5. Physical Approach to Deriving Iterative Methods -- 3.5.1. First-Order Methods -- 3.5.2. Accelerated Approach: Dynamic Relaxation -- 3.6. Multigrid Methods -- 3.7. Krylov Space Methods -- 3.7.1. Steepest Descent and Conjugate Gradient -- 3.7.2. Generalized Minimum Residual (GMRES) -- 3.7.3. Preconditioning -- 3.8. Finite Fourier Transform -- 3.8.1. Fast Fourier Transform -- 3.8.2. Application to 2D Elliptic Equations -- 3.9. Summary -- 4. Plasma Equilibrium -- 4.1. Introduction -- 4.2. Derivation of the Grad-Shafranov Equation -- 4.2.1. Equilibrium with Toroidal Flow -- 4.2.2. Tensor Pressure Equilibrium -- 4.3. Meaning of Ψ -- 4.4. Exact Solutions -- 4.4.1. Vacuum Solution -- 4.4.2. Shafranov-Solovev Solution -- 4.5. Variational Forms of the Equilibrium Equation -- 4.6. Free Boundary Grad-Shafranov Equation -- 4.6.1. Inverting the Elliptic Operator -- 4.6.2. Iterating on Jφ(R, Ψ) -- 4.6.3. Determining Ψ on the Boundary -- 4.6.4. Von Hagenow's Method -- 4.6.5. Calculation of the Critical Points -- 4.6.6. Magnetic Feedback Systems -- 4.6.7. Summary of Numerical Solution -- 4.7. Experimental Equilibrium Reconstruction -- 4.8. Summary -- 5. Magnetic Flux Coordinates in a Torus -- 5.1. Introduction -- 5.2. Preliminaries -- 5.2.1. Jacobian -- 5.2.2. Basis Vectors -- 5.2.3. Grad, Div, Curl -- 5.2.4. Metric Tensor -- 5.2.5. Metric Elements -- 5.3. Magnetic Field, Current, and Surface Functions -- 5.4. Constructing Flux Coordinates from Ψ(R, Z) -- 5.4.1. Axisymmetric Straight Field Line Coordinates -- 5.4.2. Generalized Straight Field Line Coordinates -- 5.5. Inverse Equilibrium Equation -- 5.5.1. q-Solver -- 5.5.2. J-Solver -- 5.5.3. Expansion Solution -- 5.5.4. Grad-Hirshman Variational Equilibrium -- 5.5.5. Steepest Descent Method -- 5.6. Summary -- 6. Diffusion and Transport in Axisymmetric Geometry -- 6.1. Introduction -- 6.2. Basic Equations and Orderings -- 6.2.1. Time-Dependent Coordinate Transformation -- 6.2.2. Evolution Equations in a Moving Frame -- 6.2.3. Evolution in Toroidal Flux Coordinates -- 6.2.4. Specifying a Transport Model -- 6.3. Equilibrium Constraint -- 6.3.1. Circuit Equations -- 6.3.2. Grad-Hogan Method -- 6.3.3. Taylor Method (Accelerated) -- 6.4. Time Scales -- 6.5. Summary -- 7. Numerical Methods for Parabolic Equations -- 7.1. Introduction -- 7.2. One-Dimensional Diffusion Equations -- 7.2.1. Scalar Methods -- 7.2.2. Non-Linear Implicit Methods -- 7.2.3. Boundary Conditions in One Dimension -- 7.2.4. Vector Forms -- 7.3. Multiple Dimensions -- 7.3.1. Explicit Methods -- 7.3.2. Fully Implicit Methods -- 7.3.3. Semi-Implicit Method -- 7.3.4. Fractional Steps or Splitting -- 7.3.5. Alternating Direction Implicit (ADI) -- 7.3.6. Douglas-Gunn Method -- 7.3.7. Anisotropic Diffusion -- 7.3.8. Hybrid DuFort-Frankel/Implicit Method -- 7.4. Summary -- 8. Methods of Ideal MHD Stability Analysis -- 8.1. Introduction -- 8.2. Basic Equations -- 8.2.1. Linearized Equations about Static Equilibrium -- 8.2.2. Methods of Stability Analysis -- 8.2.3. Self-Adjointness of F -- 8.2.4. Spectral Properties of F -- 8.2.5. Linearized Equations with Equilibrium Flow -- 8.3. Variational Forms -- 8.3.1. Rayleigh Variational Principle -- 8.3.2. Energy Principle -- 8.3.3. Proof of the Energy Principle -- 8.3.4. Extended Energy Principle -- 8.3.5. Useful Identities -- 8.3.6. Physical Significance of Terms in δWf -- 8.3.7. Comparison Theorem -- 8.4. Cylindrical Geometry -- 8.4.1. Eigenmode Equations and Continuous Spectra -- 8.4.2. Vacuum Solution -- 8.4.3. Reduction of δWf -- 8.5. Toroidal Geometry -- 8.5.1. Eigenmode Equations and Continuous Spectra -- 8.5.2. Vacuum Solution -- 8.5.3. Global Mode Reduction in Toroidal Geometry -- 8.5.4. Ballooning Modes -- 8.6. Summary -- 9. Numerical Methods for Hyperbolic Equations -- 9.1. Introduction -- 9.2. Explicit Centered-Space Methods -- 9.2.1. Lax-Friedrichs Method -- 9.2.2. Lax-Wendroff Methods -- 9.2.3. MacCormack Differencing -- 9.2.4. Leapfrog Method -- 9.2.5. Trapezoidal Leapfrog -- 9.3. Explicit Upwind Differencing -- 9.3.1. Beam-Warming Upwind Method -- 9.3.2. Upwind Methods for Systems of Equations -- 9.4. Limiter Methods -- 9.5. Implicit Methods -- 9.5.1. θ-Implicit Method -- 9.5.2. Alternating Direction Implicit (ADI) -- 9.5.3. Partially Implicit 2D MHD -- 9.5.4. Reduced MHD -- 9.5.5. Method of Differential Approximation -- 9.5.6. Semi-Implicit Method -- 9.5.7. Jacobian-Free Newton-Krylov Method -- 9.6. Summary -- 10. Spectral Methods for Initial Value Problems -- 10.1. Introduction -- 10.1.1. Evolution Equation Example -- 10.1.2. Classification -- 10.2. Orthogonal Expansion Functions -- 10.2.1. Continuous Fourier Expansion -- 10.2.2. Discrete Fourier Expansion -- 10.2.3. Chebyshev Polynomials in ( -1, 1) -- 10.2.4. Discrete Chebyshev Series -- 10.3. Non-Linear Problems -- 10.3.1. Fourier Galerkin -- 10.3.2. Fourier Collocation -- 10.3.3. Chebyshev Tau -- 10.4. Time Discretization -- 10.5. Implicit Example: Gyrofluid Magnetic Reconnection -- 10.6. Summary -- 11. Finite Element Method -- 11.1. Introduction -- 11.2. Ritz Method in One Dimension -- 11.2.1. Example -- 11.2.2. Linear Elements -- 11.2.3. Some Definitions -- 11.2.4. Error with Ritz Method -- 11.2.5. Hermite Cubic Elements -- 11.2.6. Cubic B-Splines -- 11.3. Galerkin Method in One Dimension -- 11.4. Finite Elements in Two Dimensions -- 11.4.1. High-Order Nodal Elements in a Quadrilateral -- 11.4.2. Spectral Elements -- 11.4.3. Triangular Elements with C1 Continuity -- 11.5. Eigenvalue Problems -- 11.5.1. Spectral Pollution -- 11.5.2. Ideal MHD Stability of a Plasma Column -- 11.5.3. Accuracy of Eigenvalue Solution -- 11.5.4. Matrix Eigenvalue Problem -- 11.6. Summary |
| 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"--Provided by publisher |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| In |
PHYSICSnetBASE |
| Subject |
Plasma (Ionized gases) -- Mathematical models
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Mathematical physics.
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SCIENCE -- Physics -- General.
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SCIENCE -- Mechanics -- General.
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SCIENCE -- Energy.
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Mathematical physics
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Plasma (Ionized gases) -- Mathematical models
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| Form |
Electronic book
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| ISBN |
9781439810958 |
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1439810958 |
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