The Hybridizable Discontinuous Galerkin Time Domain Method to Solve the 3D Maxwell's Equations

The hybridizable discontinuous Galerkin (HDG) method is very popular at present because of its lower number of globally coupled degrees of freedom compared with the discontinuous Galerkin (DG) method. However there is pretty few research about HDG in time domain computational electromagnetics. Considering the time domain methods can contain more the transient information and reflect directly the electromagnetic response characteristics compared the frequency domain, we propose a new hybridizable discontinuous Galerkin time domain (HDGTD) method to solve the Three-Dimensional (3D) time domain Maxwell's equations in this paper. We use the implicit time intergration to achieve unconditional stability compared with the explicit DGTD, which obtains bigger time step and fewer the freedom of unknown. Applying the HDGTD method to numerical example, we further verify its Feasibility and accuracy compared with the DGTD method and the analytical solutions.

I. Introduction

The discontinuous Galerkin (DG) method [1]–[3] has been a popular research area in recent years. Since 2002, the DG method has been applied in computational electromagnetics. With the development of the DG, there are more and more studies and achievements [2]–[3]. However, the DG method has an obvious disadvantage. When dealing with static problems, such as the time-harmonic Max-well's equations, the number of globally coupled degrees of freedom will be asked for more than classical the Finite Element Method (FEM) [4] if asked for same precision, which leads more computing time and memory consuming. In order to solve this issue, hybridizable discontinuous Galerkin (HDG) method [5] has been developed out of the DG during the frequency domain in recent years. The HDG method introduces a conservativity condition at the interface between neighbouring elements of the underlying discretization mesh, and numerical flux can be redefined by the additional term named hybrid variable [5]. Therefore, the HDG method can generate a linear system that only related to hybrid variable, which reduce the global coupled unknowns to the approximate trace of the solution on boundaries. There are lots of research about the HDG method, including its convergence properties [6], but it is pretty few about time domain. Especially time domain methods are popular and efficient in computational electromagnetics at present because it can contain more the transient information and reflect directly the electromagnetic response characteristics.