Abstract
An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, when combined with an explicit integration method to numerically solve a time-dependent partial differential equation, this readily leads to unduly large step size restrictions caused by the smallest grid elements. If the local refinement is strongly localized such that the ratio of fine to coarse elements is small, the unduly step size restrictions can be overcome by blending an implicit and an explicit scheme where only solution variables living at fine elements are implicitly treated. 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. We propose to present two locally implicit methods for the time-domain Maxwell’s equations. Our purpose is to compare the two with DG spatial discretization so that the most efficient one can be advocated for future use. Finally we will present a preliminary numerical investigation to increase the order of convergence.
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Notes
- 1.
Hh (t) and E h (t) denote the exact solutions of the Maxwell problem under consideration, restricted to the space grid.
- 2.
The a priori convergence analysis for this DGTD method based on a centered numerical flux, formulated on simplicial meshes, provides a convergence rate in \(\mathcal{O}({h}^{p})\) for a p-th interpolation order.
- 3.
We focus on the convergence order and not on the practical virtues of locally implicit methods. In the latter case it would be more appropriate to consider non-conforming meshes with a local refinement.
- 4.
We have also conduct the numerical test for the fully explicit case, i.e. with (5) as the basis method, and there was no reduction order for each fourth-order technique and this particular test case (no source term, no dissipative term).
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Acknowledgements
The author wishes to acknowledge the many and important contributions to this work by Stéphane Descombes, Stéphane Lanteri and Jan Verwer.
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Moya, L. (2013). Locally Implicit Discontinuous Galerkin Methods for Time-Domain Maxwell’s Equations. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_14
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DOI: https://doi.org/10.1007/978-3-642-33134-3_14
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