Stochastic multisymplectic PDEs and their structure-preserving numerical methods

TL;DR

This paper extends multisymplectic PDEs to stochastic systems using a variational approach, establishing conservation laws and developing structure-preserving numerical methods validated on stochastic nonlinear Schrödinger equations.

Contribution

It introduces a stochastic variational framework for multisymplectic PDEs, deriving conservation laws and designing structure-preserving collocation methods.

Findings

Methods preserve stochastic 2-form conservation laws

Linear systems' momentum is conserved discretely

Effective in simulating stochastic nonlinear Schrödinger equations

Abstract

We construct stochastic multisymplectic systems by considering a stochastic extension to the variational formulation of multisymplectic partial differential equations proposed in [Hydon, {\it Proc. R. Soc. A}, 461, 1627--1637, 2005]. The stochastic variational principle implies the existence of stochastic $1$-form and $2$-form conservation laws, as well as conservation laws arising from continuous variational symmetries via a stochastic Noether's theorem. These results are the stochastic analogues of those found in deterministic variational principles. Furthermore, we develop stochastic structure-preserving collocation methods for this class of stochastic multisymplectic systems. These integrators possess a discrete analogue of the stochastic $2$-form conservation law and, in the case of linear systems, also guarantee discrete momentum conservation. The effectiveness of the proposed methods is demonstrated through their application to stochastic nonlinear Schr\"odinger equations featuring either stochastic transport or stochastic dispersion.