A technique for studying strong and weak local errors of splitting stochastic integrators
Provides a new analytical tool for researchers analyzing splitting methods for stochastic differential equations, though the improvement is incremental over existing B-series approaches.
The paper introduces a word series technique to systematically derive expansions of strong and weak local errors for splitting stochastic integrators, leading to order conditions. It is applied to compare two Langevin integrators, clarifying why one outperforms the other.
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.