Arbitrary High-order Discontinuous Galerkin Method for Electromagnetic Field Problems

In this paper, we present a time integration scheme applied to the Discontinuous Galerkin finite element method (DG-FEM, [1]) for the computation of electromagnetic fields in the interior of three-dimensional structures. This approach is also known as Arbitrary High-Order Derivative Discontinuous Galerkin (ADER-DG, [2, 3]). By this method, we reach arbitrary high accuracy not only in space but also in time. The DG-FEM allows for explicit formulations in time domain on unstructured meshes with high polynomial approximation order. Furthermore, the Discontinuous Galerkin method in combination with the arbitrary high order time integration scheme is well suited to be used on massively parallel computing architectures. Moreover the method can be extended for local time stepping to become more efficiently by reducing the computation time [4]. INTRODUCTION For the design and optimization of Higher-Order-Mode Coupler (Fig. 1), used in RF accelerator structures, numerical computations of electromagnetic fields as well as scattering parameter are essential. These computations can be carried out in time domain. In this work the implementation and investigation of a time integration scheme based on the Discontinuous Galerkin Finite-Element Method (DGFigure 1: Tapered beam pipe with Higher-Order-Mode Coupler. The aspect ratio of such a grid is relatively high, due to the filigree structure of the antenna. Therefore, such a model is well suited for the application of local time stepping schemes. Work supported by Federal Ministry for Research and Education BMBF under contract 05K10HRC † [email protected] FEM) with arbitrary order in space and time is demonstrated for solving 3-D electromagnetic problems in time domain. THE NUMERICAL SCHEME With known initially and boundary conditions it is sufficient to describe classic electromagnetic phenomena only by AMPERE’s and FARADAY’s law of MAXWELL equations. The partial differential equations can be written in the following general form: ∂u ∂t +A1 ∂u ∂x +A2 ∂u ∂y +A3 ∂u ∂z = 0, (1) where u(x, y, z, t) = (Ex, Ey, Ez, Hx, Hy, Hz) T . (2) The space-dependent Jacobian matrices Ai determine the physical behavior of the equations. They contain material properties as well as the curl operator applied to Eand Hfield. To solve this ordinary partial differential equation, a physical initial condition as well as boundary conditions are still needed. For the numerical scheme the computational domain Ω ∈ R will be partitioned into conforming tetrahedral elements D. The approximate solution uh of (1) inside each tetrahedronD is given by: