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

In this paper we study the approximation by splitting techniques of the ordinary differential equation U+A U+B U=0, U(0)=U0 with A and B two matrices. We assume that we have a stiff problem in the sense that A is ill-conditionned and U0 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 U0 and the commutator between A and B.