Analysis of Approximate Factorization in Iteration Methods. Modelling, Analysis211 and Simulation

摘要

The authors consider the systems of ordinary differential equations (ODEs)211u001eobtained by spatial discretization of multi-dimensional partial differential 211u001eequations. In order to solve the initial value problem (IVP) for such ODE systems 211u001enumerically, we need a stiff IVP solver, because the Lipschitz constant 211u001eassociated with the righthand side function f becomes increasingly large as the 211u001espatial resolution is refined. Stiff IVP solvers are necessarily implicit, so 211u001ethat we are faced with the problem of solving large systems of implicit 211u001erelations. In the solution process of the implicit relations one may exploit the 211u001efact that the righthand side function f can often be split into functions fi 211u001ewhich contain only the discretizations of derivatives with respect to one spatial 211u001edimension. In this paper, the authors analyze iterative solution methods based on 211u001eapproximate factorization which are suitable for implementation on parallel 211u001ecomputer systems. In particular, the authors derive convergence and stability 211u001eregions.