Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise

TL;DR

This paper develops high-order stochastic symplectic Runge-Kutta methods for Hamiltonian systems with additive noise, ensuring accurate long-term simulation and preserving key properties like linear growth.

Contribution

It introduces new high-order symplectic schemes combining mean-square order and symplectic conditions without derivatives, verified through numerical experiments.

Findings

Achieved mean-square order 1.5 and 2.0 schemes for Hamiltonian systems

Numerical results confirm convergence order and long-term property preservation

Linear growth property maintained exactly in stochastic harmonic oscillator simulations

Abstract

In this paper, we construct stochastic symplectic Runge--Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes. We also achieve mean-square order 2.0 symplectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis. Finally, linear and non-linear systems are solved numerically, which verifies the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, the linear growth property can be preserved exactly over long-time simulation.