Convergence analysis of the Gauss-Seidel preconditioner for discretized one dimensional Euler equations

摘要

We consider the nonlinear system of equations that results from the Van Leer flux vector-splitting discretization of the one dimensional Euler equations. This nonlinear system is linearized at the discrete solution. The main topic of this paper is a convergence analysis of block-Gauss-Seidel methods applied to this linear system of equations. Both the lexicographic and the symmetric block-Gauss-Seidel method are considered. We derive results which quantify the quality of these methods as preconditioners. These results show, for example, that for the subsonic case the symmetric Gauss-Seidel method can be expected to be a much better preconditioner than the lexicographic variant. Sharp bounds for the condition number of the preconditioned matrix are derived. [References: 28]