Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes

Abstract

Abstract. A Discontinuous Galerkin method is used for to the numerical solution of the time-domainMaxwell equations on unstructured meshes. The method relies on the choice of local basis functions, a centered mean approximation for the surface integrals and a second-order leap-frog scheme for advancing in time. The method is proved to be stable for cases with either metallic or absorbing boundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved for metallic cavities. Convergence is proved for P k Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property. Promising numerical examples with low-order elements show the potential of the method.

Recent Achievements on a DGTD Method for Time-Domain Electromagnetics

“…More precisely, we set E k|a ik = − E i|a ik and H k|a ik = H i|a ik . A similar approach is applied to the numerical treatment of the absorbing boundary condition which is taken into account through the use of a fully upwind numerical flux for the evaluation of the corresponding boundary integral over a ik ∈ Γ a (see [3] for more details). Let us denote by E i and H i respectively the column vectors (E il ) 1≤l≤di and (H il ) 1≤l≤di .…”

“…In [3], the semidiscrete system (6) is time integrated using a second-order Leap-Frog scheme and it is proved that the resulting DGTD-P pi method is stable under the CFL-like condition.…”

“…In the recent years, there has been an increasing interest in DGTD methods which have been developed on quadrangular/hexahedral [1] and triangular/tetrahedral [2] meshes. In this paper, we report on some recent achievements for improving the accuracy and the computational efficiency of a DGTD method that was originally introduced in [3]. The topics addressed here are concerned with (a) dealing with a non-conforming local refinement of the mesh and a local definition of the approximation order, (b) designing a hybrid explicit-implicit time stepping strategy and, (c) computing on curvilinear domains.…”

Elastic Wave Propagation

“…For the time discretization, we apply an explicit leap-frog scheme ( [5], [1], [3]) which results, when combined with the flux (5), in a non-dissipative scheme [3]…”

“…Neglected during twenty years, it became very popular to solve hyperbolic problems especially in computational electromagnetics [5]. In spite of its succes in many domains of applications, this method has been rarely applied to seismic wave propagation problems.…”