Efficient Stochastic Galerkin Methods for Maxwell's Equations with Random Inputs
Efficient Stochastic Galerkin Methods for Maxwell's Equations with Random Inputs
复制标题
具有随机输入的麦克斯韦方程组的高效随机伽辽金方法
DOI：
10.1007/s10915-019-00936-z
复制
发表时间：
2019
影响因子：
2.5
通讯作者：
Zhou Tao
中科院分区：
文献类型：
--
作者：
Fang Zhiwei;Li Jichun;Tang Tao;Zhou Tao
本文研究了具有随机输入的含时麦克斯韦方程组的随机Galerkin方法。首先采用广义多项式混沌方法将原随机麦克斯韦方程转化为确定性展开系数方程组（Galerkin系统）。结果表明，随机伽辽金方法保持能量守恒定律。然后，我们提出了一个有限元方法在物理空间中解决的Galerkin系统，并给出了误差估计。对于时域方法，我们提出了两种离散格式，即Crank-Nicolson格式和蛙跳格式。对于Crank-Nicolson格式，我们证明了全离散格式的能量守恒性质。而对于经典的蛙跳格式，我们给出了一个条件能量稳定性性质。众所周知，对于随机Galerkin方法，主要的挑战是如何有效地解决耦合Galerkin系统。为此，我们设计了一个修改后的蛙跳型计划，其中一个可以解决耦合系统的解耦方式产生一个非常有效的数值方法。数值例子来支持理论发现。
In this paper, we are concerned with the stochastic Galerkin methods for time-dependent Maxwell’s equations with random input. The generalized polynomial chaos approach is first adopted to convert the original random Maxwell’s equation into a system of deterministic equations for the expansion coefficients (the Galerkin system). It is shown that the stochastic Galerkin approach preserves the energy conservation law. Then, we propose a finite element approach in the physical space to solve the Galerkin system, and error estimates is presented. For the time domain approach, we propose two discrete schemes, namely, the Crank–Nicolson scheme and the leap-frog type scheme. For the Crank–Nicolson scheme, we show the energy preserving property for the fully discrete scheme. While for the classic leap-frog scheme, we present a conditional energy stability property. It is well known that for the stochastic Galerkin approach, the main challenge is how to efficiently solve the coupled Galerkin system. To this end, we design a modified leap-frog type scheme in which one can solve the coupled system in a decouple way—yielding a very efficient numerical approach. Numerical examples are presented to support the theoretical finding.