Strang's formula for holomorphic semi-groups

Abstract

Let E(t)=exp(−t(A+B)) and let W(t) be the Strang approximation to E(t): W(t)=exp(−tA/2)exp(−tB)exp(−tA/2). In this article, we give a formal Taylor expansion with remainder for W(t), where the derivation operator is replaced by the bracket with one of the operators A or B. We validate this expansion in several functional cases where A and B generate a holomorphic semi-group. They include the case of the Kač transfer operator, and the case A=−MΔ with M a non-necessarily diagonal matrix with spectrum included in {Rz>0} and B the multiplication by a spatially dependent matrix. We infer stability estimates and estimates on ∥E(t)−W(t/n)n∥ when n tends to infinity.