In this article, penalty factor threshold and time step bound in discontinuous Galerkin time method based on vector wave equation (DGTD-WE) method are well estimated. Based on the semidiscrete form of the DGTD-WE method, properties of the system matrices are studied and the stability condition related to the penalty factor is derived. By decomposing the global system matrices of the DGTD-WE method into the local ones and developing an efficient iteration procedure, the lower bound threshold of the penalty factor is well estimated to guarantee the positive semidefinite property of the global system matrices. With the calculated penalty factor, the maximum time step is analytically determined by approximating spectral radius of the local system matrix. Both the penalty factor bound and the maximal time step are computed element-wise instead of a global system matrix operation, and thus, the proposed method can be efficiently applied into the large-scale meshes with different types of the basis functions and boundary conditions. Numerical examples are presented to demonstrate the validity and good performance of the proposed methods.

中文翻译：

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基于亥姆霍兹方程的不连续伽辽金时域方法的惩罚因子阈值和时间步长估计

本文对基于矢量波动方程的不连续伽辽金时间法(DGTD-WE)中的惩罚因子阈值和时间步界进行了很好的估计。基于DGTD-WE方法的半离散形式，研究了系统矩阵的性质，推导出与惩罚因子相关的稳定性条件。通过将DGTD-WE方法的全局系统矩阵分解为局部系统矩阵并开发有效的迭代程序，可以很好地估计惩罚因子的下界阈值，以保证全局系统矩阵的半正定性质。使用计算出的惩罚因子，最大时间步长是通过近似本地系统矩阵的谱半径来分析确定的。惩罚因子边界和最大时间步长都是逐元素计算的，而不是全局系统矩阵运算，因此，所提出的方法可以有效地应用于具有不同类型基函数和边界条件的大规模网格。数值例子证明了所提出方法的有效性和良好的性能。