By Frances Bauer
During this booklet, we record on learn in equipment of computational magneto hydrodynamics supported through the U.S. division of power lower than agreement EY-76-C-02-3077 with long island college. The paintings has re sulted in a working laptop or computer code for mathematical research of the equilibrium and balance of a plasma in 3 dimensions with toroidal geometry yet no sym metry. The code is indexed within the ultimate bankruptcy. models of it were used for the layout of experiments on the Los Alamos medical Laboratory and the Max Planck Institute for Plasma Physics in Garching. we're thankful to Daniel Barnes, Jeremiah Brackbill, Harold Grad, William Grossmann, Abraham Kadish, Peter Lax, Guthrie Miller, Arnulf Schliiter, and Harold Weitzner for plenty of valuable discussions of the speculation. we're specifically indebted to Franz Herrnegger for theoretical and pedagogical reviews. Constance Engle has supplied extraordinary counsel with the typescript. We enjoy acknowledging the aid of the employees of the Courant arithmetic and Com puting Laboratory at manhattan college. particularly we should always wish to exhibit our due to Max Goldstein, Kevin McAuliffe, Terry Moore, Toshi Nagano and Tsun Tam. Frances Bauer long island Octavio Betancourt September 1978 Paul Garabedian v Contents bankruptcy 1. advent 1 1. 1 formula of the matter 1 1. 2 dialogue of effects 2 bankruptcy 2. The Variational precept four four 2. 1 The Magnetostatic Equations 6 2. 2 Flux Constraints within the Plasma . 7 2. three The Ergodic Constraint .
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Additional resources for A Computational Method in Plasma Physics
Of a discrete approximation of Laplace's equation AcjJ = 0 in one dimension with the boundary conditions cjJ(O) = a, cjJ(1) = b. The simplest iterative . scheme is given by cjJ't 1 = HcjJj + 1 + cjJj - 1] = cjJj + L( cjJj). 34 3 The Discrete Equations If we divide by h 2 /2 and set Ilt = h 2 /2, this relation can be interpreted as a finite difference approximation to the heat equation ¢t = ¢xx on the lattice x = jh, t = n Ilt. Such a procedure corresponds to the method of steepest descent for minimization of Dirichlet's integral J ¢~ dx.
28 particular, it is easier to compute an m = 1, k = 1 instability than an m = 2, k = 1 instability. For the latter, the number of mesh points in the u direction is most important. Only high f3 cases are found to be unstable by the code for the kind of mesh sizes we are able to use. The accuracy of the equilibrium computation can be checked by considering various exact solutions and seeing how well the code reproduces them. This has been done for several different versions of the screw pinch, and the results are satisfactory.
Fig. 7 Comparison of computed and exact growth rates. 5 Applications 56 ARTIFICIAL TIME = a . 28 A. Fig. 8 Flux surfaces for m = 2 instability at two different times. 8. 62. ARTIFICIAL TIME =3 =1. 9. 28 particular, it is easier to compute an m = 1, k = 1 instability than an m = 2, k = 1 instability. For the latter, the number of mesh points in the u direction is most important. Only high f3 cases are found to be unstable by the code for the kind of mesh sizes we are able to use. The accuracy of the equilibrium computation can be checked by considering various exact solutions and seeing how well the code reproduces them.
A Computational Method in Plasma Physics by Frances Bauer