The Application Of Time-domain Discontinuous Galerkin Method In Computational Electromagnetics

In this paper, we studied the fundamental theory of Time-Domain Discontinuous Galerkin Method(DGTD), and combined it with Maxwell Equation in 1-D and 2-D. It mainly includes three parts: 1) the discussion of the fundamental theory of DGTD and its key idea — numerical fluxes; 2) two different temporal integration schemes and the implementation of various electromagnetic boundary conditions; 3) numerical model of the Debye dispersive medium with DGTD in 2-D.The thesis mainly consists of the following ideas:1. We introduced the DGTD theory and compared it with other electromagnetic numerical methods. We gave the weak form of DGTD system based on the conservation form of Maxwell Equations. In the introduction of DGTD weak form, the spatial distribution of the higher order nodal basis function is presented which is based on the Legendre-Gauss-Lobatto points. Then we presented and discussed the numerical properties of two different numerical fluxes, including central flux and upwind flux, and gave the discrete linear system of DGTD based on the two fluxes. After obtaining the discrete system, different temporal integration schemes—the Runge-Kutta method and Leap Frog method—are introduced and discussed to solve the ordinary differential equation system. Finally, the numerical example shows the hp-adaptivity of the DGTD method.2. The DGTD system in 2-D is presented, including the equation of spatial distribution of the nodal basis function, the weak form and the numerical fluxes in 2D. Then we discussed the implementation of various electromagnetic boundary conditions, e.g. PEC, PMC, etc. Moreover, the total-field/scatter field boundary condition and the 1st order Silver-Müller absorbing condition are adopted to simulate the scattering and radiation problems. However, in order to get a good absorbing effect of electromagnetic waves, we incorporated the Uniaxial Perfectly Matched Layer(UPML) with DGTD. The Auxiliary Equation Method(ADE) is used to model the UPML in DGTD. Similarly, the same technique is employed to obtain the discrete system of Debye medium in DGTD. An improved temporal integration scheme based on Runge-Kutta method is adopted to solve the Debye equations. The numerical examples which are compared with the analytical results including FEM and FDTD demonstrated the validity of the method.