Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations
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Abstract:
This paper is dedicated to the full discretization of linear Maxwell’s equations, where the space discretization is carried out with a discontinuous Galerkin (dG) method on a locally refined spatial grid. For such problems explicit time integrators are inefficient due to their strict CFL condition stemming from the fine grid elements. In the last few years this issue of so-called grid-induced stiffness was successfully tackled with locally implicit time integrators. So far, these methods are limited to unstabilized (central fluxes) dG methods. However, stabilized (upwind fluxes) dG schemes provide many benefits and thus are a popular choice in applications. In this paper we construct a new variant of a locally implicit time integrator using an upwind fluxes dG discretization on the coarse part of the grid. The construction is based on a rigorous error analysis which shows that the stabilization operators have to be split differently than the Maxwell operator. Moreover, our earlier analysis of a central fluxes locally implicit method based on semigroup theory applies but does not yield optimal convergence rates. In this paper we rigorously prove the stability and provide error bounds of the new method with optimal rates in space and time by means of an energy technique for a suitably defined modified error.References
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Additional Information
- Marlis Hochbruck
- Affiliation: Institute for Applied and Numerical Analysis, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany
- MR Author ID: 332872
- Email: marlis.hochbruck@kit.edu
- Andreas Sturm
- Affiliation: Institute for Applied and Numerical Analysis, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany
- MR Author ID: 1183900
- Email: andreas.sturm@kit.edu
- Received by editor(s): May 26, 2017
- Received by editor(s) in revised form: December 29, 2017, and January 25, 2018
- Published electronically: June 19, 2018
- Additional Notes: We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1121-1153
- MSC (2010): Primary 65M12, 65M15; Secondary 65M60, 65J10
- DOI: https://doi.org/10.1090/mcom/3365
- MathSciNet review: 3904141