Error analysis of a first-order DoD cut cell method for 2D unsteady advection

TL;DR

This paper presents an error analysis of a first-order cut cell method with DoD stabilization for 2D unsteady advection, enabling stable explicit time stepping despite small cut cells.

Contribution

It introduces a novel error analysis framework for a stabilized DG scheme on cut cell meshes, proving quasi-optimal error bounds for the method.

Findings

Error bounds of order one-half in combined norms.

Stability of explicit schemes on small cut cells.

Numerical validation of theoretical results.

Abstract

In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the $L^\infty$-in-time $L^2$-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results.