Derivation of optimal stochastic Runge-Kutta methods with exotic and decorated Butcher series for the weak integration of stochastic dynamics

TL;DR

This paper introduces a novel algebraic approach for designing optimal stochastic Runge-Kutta methods with high weak order, reducing computational complexity and improving efficiency in stochastic dynamics simulations.

Contribution

It presents a new method leveraging specific random Runge-Kutta coefficients to simplify order condition analysis and develop efficient second-order weak integrators.

Findings

Developed stochastic Runge-Kutta methods with optimal function evaluations

Reduced the number of random variables needed for weak order 2 methods

Confirmed efficiency improvements through numerical experiments

Abstract

The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong approximations and adapt them for the weak approximation by replacing the iterated stochastic integrals by appropriate random variables. The methods obtained this way are sub-optimal in their number of function evaluations and the analysis of order conditions is unnecessarily complicated. We provide in this paper a novel approach, relying on well-chosen sets of random Runge-Kutta coefficients, that greatly reduce the number of order conditions. The approach is successfully applied to the creation of a collection of new stochastic Runge-Kutta methods of second weak order with an optimal number of function evaluations and a smaller number of random variables. The efficiency of the new methods is confirmed with numerical experiments and a modern algebraic approach using Hopf algebras is provided for the derivation and the study of the order conditions.