非线性非一致介质二维Maxwell方程leap-frog Crank-Nicolson多区域Legendre-tau配置谱方法

摘要/Abstract

摘要：

该文研究了具有非线性电导率的非一致介质二维Maxwell方程的数值求解方法, 提出了多区域Legendre-tau配置谱方法. 该数值方法空间上达到谱精度, 时间上是二阶精度. 时间方向采用leap-frog Crank-Nicolson三层格式进行离散. 非线性项放在中间已知层采用谱配置法显式处理, 线性项采用Legendre谱方法隐式处理. 利用显隐数值格式, 既有较好的稳定性又方便算法实施. 基于合理的弱形式,不需要使用额外附加的连接条件, 以自然边界条件的方式处理交界面条件. 定义不同次数多项式逼近空间, 构建一致的数值格式. 详细证明了半离散和全离散数值格式的稳定性和收敛性, 并得到$L^2$ -范数的最优误差估计. 算例中, 利用快速Legendre变换在Chebyshev点上计算非线性项, 提高算法效率. 数值结果证实了该数值方法求解此类非线性问题的有效性, 并且没有因为解的间断而损失谱精度.

关键词: 二维Maxwell方程, 非线性电导率, 非一致介质, 多区域Legendre-tau谱方法, leap-frog Crank-Nicolson 方法, 最优误差估计

Abstract:

In this paper, numerical methods for solving 2D nonlinear Maxwell's equations in inhomogeneous media are discussed. A multidomain Legendre-tau collocation spectral method is proposed. The proposed method is of spectral accuracy in space and second order in time. In time direction, the leap-frog Crank-Nicolson method is applied, which is a three-level scheme with the nonlinear term being treated by some collocation methods explicitly in intermediate level and the linear terms being treated by the Legendre-tau spectral method implicitly. By the implicit-explicit scheme, the numerical method is of better stability and easy implementation. We construct a reasonable weak form which can deal with the interface conditions in a way just like the natural boundary condition without any additional interface conditions. The uniform scheme without any additional interface conditions is constructed by using polynomial spaces of different degrees. For the semi-discrete and fully discrete schemes, the stability and convergence are proved, and the $L^2$-norm optimal error estimates are obtained. In numerical examples, the nonlinear terms are computed at the Chebyshev points by the fast Legendre transform to improve the efficiency of the algorithm. Numerical results show the efficiency of the proposed method for solving this nonlinear problem. Moreover, the results indicate that the spectral accuracy is achieved and not affected by the discontinuity of solutions.

Key words: 2D Maxwell's equations, Nonlinear conductivity, Inhomogeneous media, Multidomain Legendre-tau spectral method, Leap-frog Crank-Nicolson method, Optimal error estimates

中图分类号:

牛翠霞,马和平. 非线性非一致介质二维Maxwell方程leap-frog Crank-Nicolson多区域Legendre-tau配置谱方法[J]. 数学物理学报, 2023, 43(3): 808-828.

Niu Cuixia,Ma Heping. A Leap-Frog Crank-Nicolson Multidomain Legendre-Tau Collocation Spectral Method for 2D Nonlinear Maxwell's Equations in Inhomogeneous Media[J]. Acta mathematica scientia,Series A, 2023, 43(3): 808-828.

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