An energy-conserving method for stochastic Maxwell equations with multiplicative noise

Abstract

In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic mutli-symplectic conservation law), and the energy of system is a conservative quantity almost surely. We propose a stochastic multi-symplectic energy-conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time. Numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy. The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated.

Stochastic multi-symplectic Runge–Kutta methods for stochastic Hamiltonian PDEs

“…Remark 3.1. The results of these two theorems are evidently consistent with the stochastic multi-symplectic conservation law and discrete energy conservation law in [13], respectively, which means that the stochastic multi-symplectic conservation law and energy conservation law can be exactly preserved by the proposed stochastic multi-symplectic RK methods (3.4)-(3.5).…”

“…Considering the following stochastic Maxwell equations with multiplicative noise in Stratonovich sense [13,18,21] dE ( In [13], the authors show that stochastic Maxwell equations can be written…”

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations

“…Such kinds of stochasticity are ubiquitous in engineering applications and are often modeled as some idealized stochastic processes, e.g., Wiener processes, Poisson processes, etc. The resulting stochastic model is termed stochastic Maxwell equations (e.g., [23,31,48]). For example, to account for the random thermal fluctuations, a stochastic external source term could be added to the Maxwell equations.…”

“…The comparison of three stochastic multi-symplectic methods, the analysis of the linear growth property of energy and the conservative property of divergence for stochastic Maxwell equations were given in [8]. Very recently, in [23], the authors constructed a stochastic multi-symplectic energy-conserving method for three-dimensional stochastic Maxwell equations with multiplicative noise using the wavelet collocation method in space and the stochastic symplectic method in time. Numerical experiments were performed to verify the excellent abilities of the proposed method.…”

Electro-rheological fluids under random influences: martingale and strong solutions

“…• The exponent p depends of the strength of the electric field which is a solution to Maxwell's equation, the latter having been widely studied in literature. Randomness naturally appears in the Maxwell equation (see, for instance [7,8] for stochastic Maxwell equations), and the randomness in the Maxwell equation transfers to randomness in the exponent in the model (1.6). In conclusion, the assumption of a random exponent is very reasonable and required by applications.…”