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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
C. Engwer, S. May, C. Nüßing, and F. Streitbürger, A stabilized discontinuous Galerkin cut cell method for discretizing the linear transport equation. arXiv:1906.05642, (2019)
E. Burman, Ghost penalty. C.R. Math., 348(21):1217–1220, (2010)
C. Gürkan and A. Massing, A stabilized cut discontinuous Galerkin framework: II. Hyperbolic problems. arXiv:1807.05634, (2018)
M. Berger, C. Helzel, and R. J. Leveque, H-box methods for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal., 41(3):893–918, (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Streitbürger, F., Engwer, C., May, S., Nüßing, A. (2021). Monotonicity Considerations for Stabilized DG Cut Cell Schemes for the Unsteady Advection Equation. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_92
Download citation
DOI: https://doi.org/10.1007/978-3-030-55874-1_92
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-55873-4
Online ISBN: 978-3-030-55874-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)