Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique, and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.
翻译:具有多角度保全法的斯多角度汉密尔顿部分差分方程式是一个重要和相当大的系统类别。 继承斯多角度汉密尔顿部分差分方程式几何特征的多视角方法为数字近似提供了更稳定的数值,对于获得正确的数值结果至关重要。 在本文中,我们提出了三种新的多视角方法,用于基于当地辐射基函数对齐法、分裂技术以及分割式龙格-库塔法的汉密尔顿部分差分方程式。 提出了非线性随机波方程式、 蒸气非线性非线性 Schr\'odinger 方程式、 STachet Korteweg-de Vries 方程式和 速性Maxwell 方程式的具体数字方法。 我们用随机波方程式作为进行数字实验的示例, 这表明拟议方法的有效性 。