A review on stochastic multi‑symplectic methods for stochastic Maxwel lequations
A review on stochastic multi‑symplectic methods for stochastic Maxwel lequations
复制标题
随机麦克斯韦方程组的随机多辛方法综述
DOI：
10.1007/s42967-019-00017-w
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发表时间：
2019
影响因子：
1.6
通讯作者：
Lihai Ji
中科院分区：
文献类型：
--
作者：
Liying Zhang;Chuchu Chen;Jialin Hong;Lihai Ji
随机多辛方法是一类保持离散随机多辛守恒律的数值方法。这些方法在求解随机Hamilton偏微分方程（PDEs）时，与传统的数值方法相比，具有显著的优越性，如长时间行为、几何结构保持和物理性质保持等。随机麦克斯韦方程组是由加性噪声或乘性噪声驱动的随机哈密顿偏微分方程组，在随机电磁学和统计辐射物理等领域有着重要的应用。因此，构造和分析随机麦克斯韦方程的各种数值方法，使其继承原系统的随机多辛性、能量和散度的演化规律，是一个重要而有前途的课题。Hong等人针对随机麦克斯韦方程组设计并分析了第一个随机多辛方法（A stochastic multi-辛scheme for stochastic麦克斯韦方程组与加性噪声. J.计算机268：255-268，2014）。随后，发展了各种随机多辛方法来求解随机麦克斯韦方程。本文对求解随机过程驱动的随机麦克斯韦方程组的随机多辛方法进行了综述。同时，也给出了随机麦克斯韦方程组的适定性和守恒律的理论结果。
Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al. (A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255–268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.