Stochastic multisymplectic PDEs and their structure-preserving numerical methods

TL;DR

This work extends the variational multisymplectic framework to stochastic PDEs by introducing a stochastic action that yields stochastic $1$-form and $2$-form conservation laws and a stochastic Noether-type theorem. It then constructs stochastic structure-preserving collocation methods (stochastic Runge–Kutta in time and space) that exactly preserve a discrete stochastic $2$-form and, in the linear/quadratic case, a discrete momentum conservation law. The approach is applied to stochastic SALT transport and stochastic nonlinear Schrödinger equations with transport and dispersion noise, and numerical experiments demonstrate preservation of key invariants and faithful soliton dynamics under noise. Overall, the paper provides both a rigorous stochastic multisymplectic theory and practical, structure-preserving numerical schemes for simulating stochastic multisymplectic PDEs.

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.

Figures (4)

• Figure 1: Evolution profile of $|\psi(x,t)|^2$ (left), the error of the global (domain integrated) conservation of wave density $|\psi(x,t)|^2$ (middle), and the error of the global conservation of wave momentum $J = \psi \nabla \psi^*$ (right) for the NLS equation with stochastic transport \ref{['eq:SALT NLS']}, driven by a single realisation of Brownian motion.
• Figure 2: The ensemble mean and standard deviation of the errors in the global conservation of wave density $|\psi(x,t)|^2$ (left) and the global conservation of wave momentum $J = \psi \nabla \psi^*$ (right) for an ensemble of $32$ members, generated by $32$ independent realisation of Brownian motion in the NLS equation with stochastic transport \ref{['eq:SALT NLS']}.
• Figure 3: Evolution profile of $|\psi(x,t)|^2$ (left), the error of the global conservation of wave density $|\psi(x,t)|^2$ (middle), and the error of the global conservation of wave momentum $J = \psi \nabla \psi^*$ (right) for the NLS equation with stochastic dispersion \ref{['eq:stoch NLS']}, driven by a single realisation of Brownian motion.
• Figure 4: The ensemble mean and standard deviation of the errors in the global conservation of wave density $|\psi(x,t)|^2$ (left) and the global conservation of wave momentum $J = \psi \nabla \psi^*$ (right) for an ensemble of $32$ members, generated by $32$ independent realisation of Brownian motion in the NLS equation with stochastic dispersion \ref{['eq:stoch NLS']}.

Theorems & Definitions (15)

• Remark 2.1: Constrained stochastic degrees of freedom
• Remark 2.2: Loss of energy conservation
• Remark 2.3
• Remark 2.4: Driving multidimensional Brownian motion
• Remark 2.5
• Remark 2.6: Notation on the expression of multisymplectic matrices
• Remark 2.7: Consistency of auxiliary variables $v$ and $w$
• Theorem 3.1
• Remark 3.1
• proof