Analysis of splitting methods for reaction-diffusion problems using stochastic calculus

Abstract

We consider linear and nonlinear reaction-diffusion problems, and their time discretization by splitting methods. We give probabilistic interpretations of the splitting schemes, and show how these representations allow us to give error bounds for the deterministic propagator under weak hypothesis on the reaction part. To show these results, we only use the Itô formula, and basic properties of solutions of stochastic differential equations. Eventually, we show how probabilistic representations of splitting schemes can be used to derive “hybrid” numerical schemes based on Monte Carlo approximations of the splitting method itself.