Penalty Factor Threshold and Time Step Bound Estimations for Discontinuous Galerkin Time-Domain Method Based on Helmholtz Equation

Abstract

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.