Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

TL;DR

This paper introduces an exotic aromatic B-series framework to analyze the accuracy of numerical integrators for ergodic SDEs, encompassing Runge-Kutta and postprocessed schemes, with a focus on invariant measures.

Contribution

It develops a novel algebraic approach using exotic aromatic B-series to systematically study integrator accuracy for ergodic SDEs.

Findings

Framework covers Runge-Kutta and partitioned methods

Exotic aromatic B-series satisfy isometric equivariance

Provides systematic accuracy analysis for invariant measures

Abstract

We introduce a new algebraic framework based on a modification (called exotic) of aromatic Butcher-series for the systematic study of the accuracy of numerical integrators for the invariant measure of a class of ergodic stochastic differential equations (SDEs) with additive noise. The proposed analysis covers Runge-Kutta type schemes including the cases of partitioned methods and postprocessed methods. We also show that the introduced exotic aromatic B-series satisfy an isometric equivariance property.