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2021 | OriginalPaper | Chapter

Monotonicity Considerations for Stabilized DG Cut Cell Schemes for the Unsteady Advection Equation

Authors : Florian Streitbürger, Christian Engwer, Sandra May, Andreas Nüßing

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2019

Publisher: Springer International Publishing

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Abstract

For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the potentially arbitrarily small cut cells as well. For explicit time stepping schemes this leads to instabilities. In a recent preprint [arXiv:1906.05642], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation. The usage of the proposed stabilization terms results in stable schemes of first and second order in one and two space dimensions. In one dimension, for piecewise constant data in space and explicit Euler in time, the stabilized scheme can even be shown to be monotone. In this contribution, we will examine the conditions for monotonicity in more detail.

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Literature
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Metadata
Title
Monotonicity Considerations for Stabilized DG Cut Cell Schemes for the Unsteady Advection Equation
Authors
Florian Streitbürger
Christian Engwer
Sandra May
Andreas Nüßing
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-55874-1_92

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