Extensions and analysis of worst-case parameter in weighted Jacobi's method for solving second order implicit PDEs

Abstract

The optimal Jacobi parameter (ω) in Jacobi's iterative method is obtained for specific classes of matrices. We define ωopt as the worst-case optimal parameter. We show that matrices with nonzero elements only along the main diagonal and odd diagonals have ωopt=1. We show ωopt→1 holds for matrices with size n and nonzero diagonal d as n,d→∞, where d is the distance from the main diagonal. Finally, we show an application which exploits these derived properties to reduce the number of required Jacobi iterations. This is especially useful for physical problems that involve 2nd order implicit PDEs (e.g. diffusion, fluids) with large sparse matrices, where a change in discretization can change which diagonals are nonzero.