ON THE LOCAL AND GLOBAL ERRORS OF SPLITTING APPROXIMATIONS OF REACTION-DIFFUSION EQUATIONS WITH HIGH SPATIAL GRADIENTS ST ´ EPHANE DESCOMBES, THIERRY DUMONT, VIOLAINE LOUVET, AND MARC MASSOT This paper is dedicated to Michel Crouzeix. Abstract. In this paper we study the approximation by splitting techniques of the ordinary differential equation ˙ U + AU + BU = 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. 1. Introduction Let M n (R) be the vector space of matrices of size n, let A and B in M n (R) and U 0 in R n . In this article we study the approximation of the following system of ordinary differential equations : (1.1)    ∂U ∂t + AU + BU =0 t> 0, U (0) = U 0 , by splitting techniques in the situation when A is ill-conditionned and the associated initial condition U 0 results in a very stiff system. The A and B matrices can be thought of as coming from the discretization of a linear partial differential equation where A corresponds to the discretization of the Laplacian operator and the initial condition, U 0 , is a vector which is the discretization of a function with a very high derivative. The exact solution of (1.1) is given by exp(−t(A + B))U 0 and is most of the time approximated when t is small enough by Lie or Strang’s formula. The Lie formula is given by (1.2) L(t)U 0 = exp (−tA) exp (−tB)U 0 , which is an approximation of local order 2 in the sense that there exist R 1 inde- pendent of t and R 2 (t) belonging to M n (R) such that for t small (1.3) e −t(A+B) U 0 − e −tA e −tB U 0 = t 2 R 1 U 0 + t 3 R 2 (t)U 0 and then is an approximation of global order 1. The Strang’s formula ([12], [13]) is given by S(t)U 0 = exp(−tA/2) exp(−tB) exp(−tA/2)U 0 . 1