A technique for studying strong and weak local errors of splitting stochastic integrators

TL;DR

This paper introduces a systematic word series technique to analyze and derive order conditions for splitting stochastic integrators, simplifying the process compared to traditional methods and clarifying the advantages of specific algorithms.

Contribution

The paper develops a new word series-based approach for analyzing local errors in splitting stochastic integrators, avoiding complex formulas like Baker-Campbell-Hausdorff.

Findings

The method provides clear order conditions for strong and weak errors.

Comparison of two Langevin integrators demonstrates the effectiveness of the approach.

Simplifies the analysis of stochastic splitting algorithms.

Abstract

We present a technique, based on so-called word series, to write down in a systematic way expansions of the strong and weak local errors of splitting algorithms for the integration of Stratonovich stochastic differential equations. Those expansions immediately lead to the corresponding order conditions. Word series are similar to, but simpler than, the B-series used to analyze Runge-Kutta and other one-step integrators. The suggested approach makes it unnecessary to use the Baker-Campbell-Hausdorff formula. As an application, we compare two splitting algorithms recently considered by Leimkuhler and Matthews to integrate the Langevin equations. The word series method bears out clearly reasons for the advantages of one algorithm over the other.