Analysis of approximate factorization in iteration methods

Abstract

We consider the systems of ordinary differential equations (ODEs) obtained by spatial discretization of multi-dimensional partial differential equations. In order to solve the initial value problem (IVP) for such ODE systems numerically, we need a stiff IVP solver, because the Lipschitz constant associated with the right-hand side function f becomes increasingly large as the spatial resolution is refined. Stiff IVP solvers are necessarily implicit, so that we are faced with the problem of solving large systems of implicit relations. In the solution process of the implicit relations one may exploit the fact that the right-hand side function f can often be split into functions f i which contain only the discretizations of derivatives with respect to one spatial dimension. In this paper, we analyze iterative solution methods based on approximate factorization which are suitable for implementation on parallel computer systems. In particular, we derive convergence and stability regions.