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The domain-of-dependence stabilization for cut-cell meshes is fully discretely stable
Louis Petri1; Gunnar Birke2; Christian Engwer2; Hendrik Ranocha1
1Institute of Mathematics, Johannes Gutenberg University Mainz, Staudingerweg 9, 55128 Mainz, Germany
2Applied Mathematics, University of Münster, Orléans-Ring 10, 48149 Münster, Germany
The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 187-218

We present a fully discrete stability analysis of the domain-of-dependence stabilization for hyperbolic problems. The method aims to address issues caused by small cut cells by redistributing mass around the neighborhood of a small cut cell at a semi-discrete level. Our analysis is conducted for the linear advection model problem in one spatial dimension. We demonstrate that fully discrete stability can be achieved under a time step restriction that does not depend on the arbitrarily small cells, using an operator norm estimate. Additionally, this analysis offers a detailed understanding of the stability mechanism and highlights some challenges associated with higher-order polynomials. We also propose a way to mitigate these issues to derive a feasible CFL-like condition. The analytical findings, as well as the proposed solution are verified numerically in one- and two-dimensional simulations.

Published online:
DOI: 10.5802/smai-jcm.147
Classification: 65M12, 65M20, 65M70
Keywords: cut-cell meshes, discontinuous Galerkin methods, domain-of-dependence stabilization, semibounded operators, energy stability, Runge–Kutta methods

Louis Petri  1 ; Gunnar Birke  2 ; Christian Engwer  2 ; Hendrik Ranocha  1

1 Institute of Mathematics, Johannes Gutenberg University Mainz, Staudingerweg 9, 55128 Mainz, Germany
2 Applied Mathematics, University of Münster, Orléans-Ring 10, 48149 Münster, Germany 
Louis Petri; Gunnar Birke; Christian Engwer; Hendrik Ranocha. The domain-of-dependence stabilization for cut-cell meshes is fully discretely stable. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 187-218. doi: 10.5802/smai-jcm.147
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