Solving the 3D MHD equilibrium equations in toroidal geometry by Newton's method
Abstract
We describe a novel form of Newton's method for computing 3D MHD equilibria. The method has been implemented as an extension to the hybrid spectral/finitedifference Princeton Iterative Equilibrium Solver (PIES) which normally uses Picard iteration on the full nonlinear MHD equilibrium equations. Computing the Newton functional derivative numerically is not feasible in a code of this type but we are able to do the calculation analytically in magnetic coordinates by considering the response of the plasma's PfirschSchlueter currents to small changes in the magnetic field. Results demonstrate a significant advantage over Picard iteration in many cases, including simple finite{beta} stellarator equilibria. The method shows promise in cases that are difficult for Picard iteration, although it is sensitive to resolution and imperfections in the magnetic coordinates, and further work is required to adapt it to the presence of magnetic islands and stochastic regions.
 Authors:
 Princeton Plasma Physics Laboratory, Princeton University, NJ (United States). Email: h.oliver@niwa.co.nz
 Princeton Plasma Physics Laboratory, Princeton University, NJ (United States). Email: reiman@pppl.gov
 Princeton Plasma Physics Laboratory, Princeton University, NJ (United States). Email: monticello@pppl.gov
 Publication Date:
 OSTI Identifier:
 20687269
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 211; Journal Issue: 1; Other Information: DOI: 10.1016/j.jcp.2005.05.007; PII: S00219991(05)002640; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COORDINATES; EQUATIONS; GEOMETRY; MAGNETIC FIELDS; MAGNETIC ISLANDS; MAGNETOHYDRODYNAMICS; MHD EQUILIBRIUM; NEWTON METHOD; NONLINEAR PROBLEMS; STOCHASTIC PROCESSES; THREEDIMENSIONAL CALCULATIONS
Citation Formats
Oliver, H.J., Reiman, A.H., and Monticello, D.A. Solving the 3D MHD equilibrium equations in toroidal geometry by Newton's method. United States: N. p., 2006.
Web. doi:10.1016/j.jcp.2005.05.007.
Oliver, H.J., Reiman, A.H., & Monticello, D.A. Solving the 3D MHD equilibrium equations in toroidal geometry by Newton's method. United States. doi:10.1016/j.jcp.2005.05.007.
Oliver, H.J., Reiman, A.H., and Monticello, D.A. Sun .
"Solving the 3D MHD equilibrium equations in toroidal geometry by Newton's method". United States.
doi:10.1016/j.jcp.2005.05.007.
@article{osti_20687269,
title = {Solving the 3D MHD equilibrium equations in toroidal geometry by Newton's method},
author = {Oliver, H.J. and Reiman, A.H. and Monticello, D.A.},
abstractNote = {We describe a novel form of Newton's method for computing 3D MHD equilibria. The method has been implemented as an extension to the hybrid spectral/finitedifference Princeton Iterative Equilibrium Solver (PIES) which normally uses Picard iteration on the full nonlinear MHD equilibrium equations. Computing the Newton functional derivative numerically is not feasible in a code of this type but we are able to do the calculation analytically in magnetic coordinates by considering the response of the plasma's PfirschSchlueter currents to small changes in the magnetic field. Results demonstrate a significant advantage over Picard iteration in many cases, including simple finite{beta} stellarator equilibria. The method shows promise in cases that are difficult for Picard iteration, although it is sensitive to resolution and imperfections in the magnetic coordinates, and further work is required to adapt it to the presence of magnetic islands and stochastic regions.},
doi = {10.1016/j.jcp.2005.05.007},
journal = {Journal of Computational Physics},
number = 1,
volume = 211,
place = {United States},
year = {Sun Jan 01 00:00:00 EST 2006},
month = {Sun Jan 01 00:00:00 EST 2006}
}

The Schurdecomposition for threedimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3Dmore »

Numerical calculations using the full MHD equations in toroidal geometry
A computer code has been constructed that solves the full magnetohydrodynamic (MHD) equations in toroidal geometry. The code is applicable to toroidal devices, including tokamaks, stellarators, and reversed field pinches. A fully implicit numerical technique is used that allows linear eigenvalues and eigenfunctions to be found in a very few computational steps. Although the present work describes the solution of the linearized equations, generalization of the numerical method to the solution of the nonlinear problem is straightforward. Use of the code is illustrated by calculating the n = 1 instability for a tokamak configuration. The results show the structural changesmore » 
Variation method for approximately solving the problem of the MHD equilibrium of a tokamak plasma
A solution to the EulerOstrogradskii equation is found for the toroidal current.(AIP) 
A fractionalstep method for solving 3D, timedomain Maxwell equations
Stable and efficient implicit and explicit fractionalstep methods for solving threedimensional, timedependent Maxwell equations have been successfully developed. These numerical procedures are characteristicbased schemes with the intrinsically accurate noreflection wave condition on the boundaries of truncated computational domain. Excellent simulations for electromagnetic phenomena have been achieved for a threedimensional wave guide and an oscillating electric dipole. 
AN ITERATIVE METHOD FOR SOLVING THE P$sub 1$ EQUATIONS IN SLAB GEOMETRY
The fewgroup diffusion codes programmed for various machines are inadequate for classes of problems requiring higher approximations to the transport equation. It is possible, however, to do onevelocity P/sub 3/ or doubleP/sub 1/ problems, in slab geometry, with the aid of a slightly modified diffusion code. This diffusion code is discussed. (A.C.)