Direct Solutions of Finite-Difference Systems for Poisson's Equation I. Simple Cases

Abstract

New solution algorithms of finite-difference systems for 1D and 2D Poisson's equations were obtained by modifying Gaussian elimination. Introduction of a new naming of row and column for coefficient matrices of the systems made it possible to derive the elimination algorithms explicitly. Since the finite-difference systems for Poisson's equations are diagonally dominant, solutions with high accuracy can be obtained with no use of pivoting in the present algorithm. The computer execution time for the present algorithm would appear to be one order smaller than the time for SOR with Chebyshev acceleration.