An approximate factorization procedure for solving nine-point elliptic difference equations Application for a fast 2-D relativistic Fokker-Planck solver

Abstract

A full implicit numerical procedure based on the use of a mine-point difference operator is presented to solve the two-dimensional (2-D) relativistic Fokker-Planck equation for the current drive problem and synergetic effects between the lower hybrid and the electron cyclotron waves in tokamaks. As compared to the standard approach based on the use of a five-point difference operator (M. Shoucri, I. Shkarofsky, Comput. Phys. Commun. 82 (1994) 287), the convergence rate towards the steady state solution may be significantly enhanced with no loss of accuracy on the distribution function. Moreover, it is shown that the numerical stability may be strongly improved without a large degradation of the CPU time consumption as in the five-point scheme, making this approach very attractive for a fast solution of the 2-D Fokker-Planck equation on a fine grid in conjunction with other numerical codes for realistic plasma simulations. This new algorithm, based on an approximate matrix factorization technique, may be applied to all numerical problems with large sets of equations which involve nine-point difference operators.