Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

TL;DR

This paper introduces modified averaged vector field methods that preserve multiple invariants in conservative stochastic differential equations, achieving mean square convergence order 1 and demonstrating superior long-term simulation performance.

Contribution

The paper proposes a new class of modified averaged vector field methods that preserve multiple invariants in stochastic differential equations, with proven convergence and improved long-term stability.

Findings

Methods preserve multiple invariants simultaneously.

Achieve mean square convergence order 1.

Numerical experiments confirm theoretical results and long-term advantages.

Abstract

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order $1$ of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation.