The domain-of-dependence stabilization for cut-cell meshes is fully discretely stable
For computational fluid dynamics, this provides a stable method for hyperbolic problems on cut-cell meshes, addressing the small cell problem without restrictive time steps.
The paper proves that the domain-of-dependence stabilization for cut-cell meshes achieves fully discrete stability under a time step restriction independent of small cells, verified by 1D and 2D simulations.
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.