A Novel Space-time Discontinuous Galerkin Method for Solving of One-dimensional Electromagnetic Wave Propagations

In this paper we propose a high-order space-time discontinuous Galerkin (STDG) method for solving of one-dimensional electromagnetic wave propagations in homogeneous medium. The STDG method uses finite element Discontinuous Galerkin discretizations in spatial and temporal domain simultaneously with high order piecewise Jacobi polynomial as the basis functions. The algebraic equations are solved using Block Gauss-Seidel iteratively in each time step. The STDG method is unconditionally stable, so the CFL number can be chosen arbitrarily. Numerical examples show that the proposed STDG method is of exponentially accuracy in time.


Introduction
The The finite element method has had a greater impact on theory and application on numerical methods during the twentieth century [1]. It has been used throughout the fields of engineering. The success of finite element method is lied on its flexibility. It can be used for solving problem with complex shape and arbitrary boundary conditions.
The achievement of the FE method is the application of the method in computational electromagnetics (CEM) remarkable, especially ini static and quasi static electromagnetics [2,3] however, the method has not produce entirely satisfactory results in time domain CEM. The governing equations of electromagnetic wave propagation are often discretized with FE method in time domian, resulting in semi algebraic equations in time, which are solved with explicit Runge Kutta family schemes or finite difference method. Explicit methods which do not require large matrix inversion were used for modeling wave propagation [4]. Difficulties of explicit methods are low order, lacking of high-frequency dissipation, and conditionally stable. In order to maintain the calculation stable, a small step size below CFL limit must be kept. Reference [5] and [6] proposed time-discontinuous Galerkin (DG) method for discretizing the temporal domain, while the spatial domain still discretized by conventional FE method. They showed that the results are very promising and the 3 th order temporal discretization can be achieved. Reference [7] developed space-time discontinuous Galerkin (STD) method for solving 1-D scalar wave propagation problems. The spatial and temporal domains are coupled strongly, 1-D wave propagation problems become 2-D problems because variable time is added as second spatial coordinate. They used unstructured triangular space-time element. They showed that (p+1) th order accuracy in L 2 norm can be achieved for arbitrary basis function order (p). While this approach having good accuracies, it has diffulties in implementation of coding and extension to real 3-D problems. Reference [8] continued the work of [5] for solving elastodynamics problems, DG method was used not only for temporal domain but spatial domain also. The elastodynamics equations are transformed into velocity-displacement formulation, so the order of equations are second order in space and first order in time. The results showed that high-order accuracy an be achieved and high-frequency oscillation can be damped.
In this paper, we propose a new space-time discontinuous Galerkin method for solving one-dimensional Maxwell's equations, which can describe electromagnetic wave propagation. The spatial and temporal domains are coupled strongly, and the high-order basis functions are tensor product of 1-dimensional Jacobi basis functions in space and time.

Governing Equations
In this paper, the 1-D system of transverse electric (TE) Maxwell's equations have been chosen.
where z H is magnetic field, y E is electric field,  is permittivity µ is permeability of medium. In this paper  and µ are kept constant and set to be equal one.
For simplicity, the governing equations are written in vector notation:

Numerical Method 3.1. Discretization
Inspired by work of Maerschalck [9], we extended the work to solve 1-dimensional STDG formulation. In the space-time formulation of equation (2), the variable t is considered as a third spatial variable. The dimension of the discretization is always one higher than dimension of the actual governing equations.
In our approach we divided the space-time domain into rectangular slabs where  is the spatial domain and  The physical space-time elements are mapped into reference elements as shown in Figure 2 using mapping function: The discrete form of equation (3) in a space-time element can be written as follows:

Results and Discussion
In this section, we demonstrate the performance of the STDG method. The following initial conditions are taken to perform numerical simulation: The exact solution for the electromagnetic fields are: We will discussed the rate of convergence and accuracy of the STDG method. We tried to examine the order of STDG formulation in time direction only, examination of the order in space direction can be found in [10]. The spatial domain is divided into 20 elements, the spatial polynomial order is kept constant N s = 4 and temporal polynomial order is varied from N t = 0 to N t = 3. We used the L 2 norm as the error indicator which is defined as:  Table 1.  The snapshots of electromagnetic fields are shown by Figures 3(a-b). Those figures described the evolution of the electromagnetic fields, actually they are standing waves. They did not propagate but oscillated.

Conclusion
We have developed a new space-time discontinuous galerkin (STDG) method for simulation of the one-dimensional electromagnetic wave fields. The new method has high accuracy, exponential orde accuracy of  