By Frances Bauer

During this booklet, we file on examine in tools of computational magneto hydrodynamics supported via the U.S. division of power below agreement EY-76-C-02-3077 with big apple collage. 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 clinical 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 lots of invaluable discussions of the idea. we're in particular indebted to Franz Herrnegger for theoretical and pedagogical reviews. Constance Engle has supplied striking counsel with the typescript. We enjoy acknowledging the aid of the employees of the Courant arithmetic and Com puting Laboratory at ny collage. specifically we must always prefer to exhibit our because of Max Goldstein, Kevin McAuliffe, Terry Moore, Toshi Nagano and Tsun Tam. Frances Bauer big apple 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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**Example text**

Been added. The coefficient e4 is selected both to meet the Courant -FriedrichsLewy stability condition and to maintain descent. It is independent of the artificial time t. 7 Remarks About the Method Because we are not only trying to solve the magnetostatic equations but also want to determine if the solution is stable, we have to distinguish between mathematical instabilities of the numerical method and physical instabilities 32 3 The Discrete Equations of the equilibrium. This is accomplished by monitoring the behavior of the energy E as a function of the artificial time parameter t.

00 E ARTIFICIAL TIME Fig. 3 Fourier coefficients and descent coefficients. ~ o CD (f) f- Z wg uo+------. ::1" lL. lL. W o Uo o ~. z~+-----.. 00 ARTIFICIAL TIME Fig. 5. It is seen that after the energy goes through an inflection point it converges to a minimum value, while the m = 1, k = 1 mode saturates. This is exactly what one would expect when a stable bifurcated solution exists [27]. Although this solution is not very remarkable physically because the plasma boundary is very near the outer wall, it does illustrate the nonlinear capability of the code.

One can easily distinguish the m = 1, k = 1 mode. As artificial time evolves the potential energy develops an inflection point, and as the solution diverges away from equilibrium the residuals as well as the amplitudes of the unstable modes increase. This is typical of unstable equilibria. 2 Comparison with Exact Solutions , N o.... ~C? 00 Fig. 2 Time behavior of energy and residuals. J cr ::::> CJ .... 0 (1)- W' a:::"? OO 8·00 RRTIFICIRL TIME 12·00 linearly with artificial time, indicating an exponential instability.