Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell's equations

Stéphane Descombes 1, 2 Stéphane Lanteri 1 Ludovic Moya 1, *
* Corresponding author
1 NACHOS - Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the ratio of fine to coarse elements is small, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The counterpart of this approach is having to solve a linear system per time step. But due to the assumed small fine to coarse elements ratio, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit-explicit approaches in presence of local refinements.
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Stéphane Descombes, Stéphane Lanteri, Ludovic Moya. Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell's equations. [Research Report] RR-7983, INRIA. 2012, pp.28. ⟨hal-00702802v2⟩

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