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, {\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 Schrödinger equations featuring either stochastic transport or stochastic dispersion.

1st version, 27 pages

References in corpus (10)

• Multisymplectic geometry, variational integrators, and nonlinear PDEs
• Variational Principles for Stochastic Fluid Dynamics
• Solution properties of a 3D stochastic Euler fluid equation
• Stochastic Variational Integrators
• Multisymplectic formulation of fluid dynamics using the inverse map
• Stochastic Discrete Hamiltonian Variational Integrators
• Momentum Maps and Stochastic Clebsch Action Principles
• Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise
• Stochastic effects of waves on currents in the ocean mixed layer
• New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map