Efficient preconditioning of linear systems arising from the discretization of hyperbolic conservation laws

Abstract

In this paper, we describe a novel formulation of a preconditioned BiCGSTAB algorithm for the solution of ill-conditioned linear systems Ax=b. The developed extension enables the control of the residual r m =b−Ax m of the approximate solution x m independent of the specific left, right or two-sided preconditioning technique considered. Thereby, the presented modification does not require any additional computational effort and can be introduced directly into existing computer codes. Furthermore, the proceeding is not restricted to the BiCGSTAB method, hence the strategy can serve as a guideline to extend similar Krylov sub-space methods in the same manner. Based on the presented algorithm, we study the behavior of different preconditioning techniques. We introduce a new physically motivated approach within an implicit finite volume scheme for the system of the Euler equations of gas dynamics which is a typical representative of hyperbolic conservation laws. Thereupon a great variety of realistic flow problems are considered in order to give reliable statements concerning the efficiency and performance of modern preconditioning techniques.