Solving procedure for a twenty-five diagonal coefficient matrix: direct numerical solutions of the three dimensional linear Fokker-Planck equation

Abstract

We describe an implicit procedure for solving linear equation systems resulting from the discretization of the three dimensional (seven variables) linear Fokker-Planck equation. The discretization of the Fokker-Planck equation is performed using a twenty-five point molecule that leads to a coefficient matrix with equal number of diagonals. The method is an extension of Stone's implicit procedure, includes a vast class of collision terms and can be applied to stationary or non stationary problems with different discretizations in time. Test calculations and comparisons with other methods are presented in two stationary examples, including an astrophysical application for the Miyamoto-Nagai disk potential for a typical galaxy.