High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface

TL;DR

This paper introduces high-order cut finite element methods within the discontinuous Galerkin framework for hyperbolic conservation laws with stationary and moving interfaces, ensuring conservation, stability, and accuracy.

Contribution

The paper develops novel high-order cut finite element methods for hyperbolic laws with interfaces, including space-time formulations for moving interfaces, with proven stability and error estimates.

Findings

Methods are conservative and energy stable.

Numerical results confirm expected accuracy.

Error estimates are established for stationary interfaces.

Abstract

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time-stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.