On the consistency of the Domain of Dependence cut cell stabilization

TL;DR

This paper proves the consistency of the Domain of Dependence stabilization method for cut cell meshes with arbitrary polynomial degrees, enabling stable explicit time stepping without small cell restrictions in hyperbolic PDEs.

Contribution

It provides the first analytical consistency proof for DoD stabilization for any polynomial degree, extending previous results limited to degree zero.

Findings

Proven consistency of DoD stabilization for arbitrary polynomial degrees.

Supports the potential for high-order accuracy in cut cell methods.

Lays groundwork for further high-order error analysis.

Abstract

So called cartesian cut cell meshes provide efficient ways to generate meshes but do require tailored numerical methods to not suffer from stabilization issues, especially in the hyperbolic regime where the application of explicit time stepping schemes is common. In this scenario, due to potentially arbitrarily small cut cells, an infeasible restriction is imposed on the time step size. The Domain of Dependence (DoD) stabilization allows for a time step size based on the underlying Cartesian mesh. Being an extension of a discontinuous Galerkin (DG) method, one would expect similar accuracy properties as in the pure DG case. While numerical results do support this expectation, on the analytical level this has only been investigated thoroughly for $k=0$. Error analysis typically hinges on a consistency result. In this contribution we prove such a result for the DoD stabilization given an arbitrary polynomial degree and an exact solution of sufficient regularity. This in turn could open the way towards a more refined analysis of the method even in the high-order case.