By Frances Bauer
During this ebook, we document on study in tools of computational magneto hydrodynamics supported through the U.S. division of strength less than agreement EY-76-C-02-3077 with manhattan 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 plenty of important discussions of the speculation. we're specially indebted to Franz Herrnegger for theoretical and pedagogical reviews. Constance Engle has supplied awesome assistance with the typescript. We enjoy acknowledging assistance from the employees of the Courant arithmetic and Com puting Laboratory at long island collage. particularly we should always prefer to show our due to Max Goldstein, Kevin McAuliffe, Terry Moore, Toshi Nagano and Tsun Tam. Frances Bauer ny Octavio Betancourt September 1978 Paul Garabedian v Contents bankruptcy 1. creation 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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Extra resources for A Computational Method in Plasma Physics
The functions r2(u, v) and ziu, v), which describe the shape of the outer wall, are known explicitly-for example, as trigonometric polynomials-so we can compute their first derivatives exactly. Then the truncation errors for the first derivatives will be bounded by derivatives of R, g, ro, and zo, which are generally smaller than those of rand z. Thus, for example, we discretize the formula r: + z: = [(rl - ro)2 + [(r1 + (ZI - rO)(rl)u - zo)2]R: + (ZI + [(rl): + (zl):]R 2 - zo)(zl)u](R 2)u instead of writing a difference approximation in terms of rand z directly.
K+! + • 1 ,J,. ',J,k+1 . ) RL+I,k+I)]. The motivation for these finite difference formulas can be understood by considering the simple case of Laplace's equation in two dimensions, which involves the Dirichlet integral Using a rectangular mesh and following the principle of squaring before averaging, we are led to the usual five-point formula for Laplace's equation. If instead we approximate the first derivatives at the center of each rectangle and then square, we are led to a five-point formula involving a mesh point together with the four points diagonally removed.
Flux surface. 3 Difference Equations for the Vacuum Region 27 computed on that flux surface. The equation is diagonally dominant because F2« EG. At the magnetic axis, where the derivatives ru and zu are equal to zero, it reduces to (%u)(Gt/lu) = 0 and can be solved explicitly using the periodicity conditions for t/I. Details of all the difference equations for the plasma can be found from the listing of the computer code in the last chapter. Therefore, we will not go into them any further here.
A Computational Method in Plasma Physics by Frances Bauer