ANALYSIS OF APPROXIMATE FACTORIZATION IN ITERATION METHODS

We consider the systems of ordinary differential equations (ODEs) obta ined by spatial discretization of multidimensional partial differentia l equations. In order to solve the initial value problem (IVP) for suc h ODE systems numerically, we need a stiff IVP solver, because the Lip schitz constant associated with the right-hand side function f becomes increasingly large as the spatial resolution is refined. Stiff IVP so lvers are necessarily implicit, so that we are faced with the problem of solving large systems of implicit relations. In the solution proces s of the implicit relations one may exploit the fact that the right-ha nd side function f can often be split into functions f(i) which contai n 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 st ability regions. (C) 1998 Elsevier Science B.V, and IMACS, All rights reserved.