The use of approximate factorization in sti ODE solvers

We consider implicit integration methods for the numerical solution of sti initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonsti or mildly sti with respect to the stepsize. In these nonsti subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly sti situations and is less costly than Newton iteration. The process we have in mind uses modi ed Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine xed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of sti ness is changing during the integration. c © 1998 Elsevier Science B.V. All rights reserved.