Preservation of Physical Properties of Stochastic Maxwell Equations with Additive Noise via Stochastic Multi-symplectic Methods
For researchers in stochastic PDEs and numerical methods, this work provides structure-preserving discretizations for a class of stochastic Hamiltonian systems.
The paper shows that stochastic Maxwell equations with additive noise preserve divergence in expectation and have linearly increasing averaged energy, and proposes three stochastic multi-symplectic methods that preserve discrete averaged divergence and match continuous energy growth rates.
Stochastic Maxwell equations with additive noise are a system of stochastic Hamiltonian partial differential equations intrinsically, possessing the stochastic multi-symplectic conservation law.It is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic Maxwell equations with additive noise preserves the divergence in the sense of expectation. Moreover, we propose three novel stochastic multi-symplectic methods to discretize stochastic Maxwell equations in order to investigate the preservation of these properties numerically. We made theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence. Meanwhile, we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method. Especially, the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case. Numerical experiments are performed to verify our theoretical results.