Convergence Analysis of the Gauss-Seidel Preconditioner for Discretized One Dimensional Euler Equations

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 blockGauss–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. AMS subject classifications. 65F10, 65N22, 65N06