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Finite Element Solution of the Fokker-Plonck Equation in Electrostatic Magnetic Mirror Fusion Machines

Taha H. Zerguini

Nuclear Science and Engineering

Volume 92 / Number 1 / January 1986 / Pages 84-91


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A perturbation method is developed to find solutions of sloshing ion distributions. This method uses an expansion in the ratio of electrostatic potential to average ion energy to simplify the bounce-averaged Fokker-Planck equation. Finite element techniques, which provide rapid numerical solutions for parametric studies of sloshing ions, are used to derive the zeroth-order angular and velocity equations. The first-order two-dimensional equation was also expanded into finite element “hat functions.” Application of Galerkin's method gives a linear system of equations where all matrix and source elements are calculated analytically. The density ratio and the potential profiles as functions of axial distance are computed. There is excellent agreement with results from the Lawrence Liver-more National Laboratory bounce-averaged Fokker-Planck code with as much as 500 times and 50 times less Cray-1 computer time for the zeroth- and the first-order solutions, respectively.

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