Stochastic Multisymplectic PDEs and Their Structure‐Preserving Numerical Methods

ABSTRACT We construct stochastic multisymplectic systems by considering a stochastic extension to the variational formulation of multisymplectic partial differential equations proposed in Hydon [ Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 461 (2005): 1627–1637]. 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 analogs 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 analog 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 Schrödinger equations featuring either stochastic transport or stochastic dispersion.