On the local and global errors of splitting approximations of reaction–diffusion equations with high spatial gradients

Abstract

In this paper we study the approximation by splitting techniques of the ordinary differential equation U˙+A U+B U=0, U(0)=U 0 with A and B two matrices. We assume that we have a stiff problem in the sense that A is ill-conditionned and U 0 is a vector which is the discretization of a function with a very high derivative. This situation may appear for example when we study the discretization of a partial differential equation. We prove some error estimates for two general matrices and in the stiff case, where the estimates are independent of U 0 and the commutator between A and B.