General order conditions for stochastic partitioned Runge-Kutta methods

TL;DR

This paper develops a comprehensive order theory for stochastic partitioned Runge-Kutta methods using stochastic B-series and rooted trees, enabling analysis and simplification of order conditions.

Contribution

It introduces a general order theory for SPRK methods based on stochastic B-series and rooted trees, and shows how to reduce order conditions in special cases.

Findings

Proves the order of some known SPRK methods.

Shows how to simplify order conditions for quadratic invariants.

Provides a framework for analyzing and designing SPRK methods.

Abstract

In this paper stochastic partitioned Runge-Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions.