High order cut discontinuous Galerkin methods for hyperbolic conservation laws in one space dimension

TL;DR

This paper introduces high order cut discontinuous Galerkin methods for 1D hyperbolic conservation laws, ensuring stability, accuracy, and TVD properties even with irregular cut elements.

Contribution

It develops a novel high order cut DG framework with ghost penalty stabilization and proves stability, accuracy, and TVD properties for the method.

Findings

Methods achieve high order accuracy on smooth problems

Stable and TVD for piecewise constant schemes

Effective for discontinuous solutions

Abstract

In this paper, we develop a family of high order cut discontinuous Galerkin (DG) methods for hyperbolic conservation laws in one space dimension. The ghost penalty stabilization is used to stabilize the scheme for small cut elements. The analysis shows that our proposed methods have similar stability and accuracy properties as the standard DG methods on a regular mesh. We also prove that the cut DG method with piecewise constants in space is total variation diminishing (TVD). We use the strong stability preserving Runge-Kutta method for time discretization and the time step is independent of the size of cut element. Numerical examples demonstrate that the cut DG methods are high order accurate for smooth problems and perform well for discontinuous problems.