An Implicit DGTD Method for Solving the Two-Dimensional Maxwell Equations on Unstructured Triangular Meshes

Abstract : Numerical methods for solving the time domain Maxwell equations often rely on cartesian meshes and are variants of the finite difference time domain method originating in the seminal work of Yee [6]. In the recent years, there has been an increasing interest in discontinuous Galerkin time domain methods dealing with unstructured meshes since the latter are particularly well suited for the discretization of geometrical details that characterize applications of practical relevance. Similarly to Yee's finite difference time domain method, discontinuous Galerkin time domain methods generally rely on explicit time integration schemes and are therefore constrained by a stability condition that can be very restrictive on highly refined or unstructured meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable. However, such a time scheme comes at the expense of the inversion of a global linear system at each time step, thus obliterating one of the attractive features of discontinuous Galerkin formulations. In this paper, we report on our recent efforts concerning the design of an implicit time integration scheme in conjunction with a discontinuous Galerkin approximation method for solving the time domain Maxwell equations on unstructured triangular meshes. Despite the memory and computational overheads induced by the inversion of a global linear system at each time step, we demonstrate that an implicit discontinuous Galerkin time domain method is a viable numerical strategy for solving electromagnetic wave propagation problems on locally refined unstructured meshes.
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Adrien Catella, Victorita Dolean, Stephane Lanteri. An Implicit DGTD Method for Solving the Two-Dimensional Maxwell Equations on Unstructured Triangular Meshes. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Emerald, 2009, To appear, pp.29. ⟨inria-00126573v4⟩

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