Stochastic Discrete Hamiltonian Variational Integrators

TL;DR

This paper develops a unified framework for deriving structure-preserving stochastic Hamiltonian integrators, including new symplectic methods, that improve long-term stability and energy conservation in simulations with multiplicative noise.

Contribution

It introduces a stochastic discrete Hamiltonian approach based on a variational principle, unifying and extending existing stochastic integrators with new symplectic schemes.

Findings

New stochastic symplectic methods with mean-square order 1.0

Methods demonstrate superior long-term stability and energy behavior

Includes stochastic symplectic Runge-Kutta as a special case

Abstract

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge-Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.