A two-dimensional stabilized discontinuous Galerkin method on curvilinear embedded boundary grids

TL;DR

This paper introduces a high-order discontinuous Galerkin method with state redistribution on curvilinear embedded boundary grids, enabling larger time steps and improved stability without complex geometric manipulations.

Contribution

The paper presents a novel state redistribution technique that relaxes CFL restrictions for DG methods on cut cell grids, improving stability and efficiency.

Findings

Converges with order p+1 in L1 norm for smooth solutions.

Capable of accurately capturing shocks.

Demonstrates potential for complex embedded geometries.

Abstract

We propose a state redistribution method for high order discontinuous Galerkin methods on curvilinear embedded boundary grids. State redistribution relaxes the overly restrictive CFL condition that results from arbitrarily small cut cells and explicit time stepping. Thus, the scheme can take time steps that are proportional to the size of cells in the background grid. The discontinuous Galerkin scheme is stabilized by postprocessing the numerical solution after each stage or step of an explicit time stepping method. This is done by temporarily merging the small cells into larger, possibly overlapping neighborhoods using a special weighted inner product. Then, the numerical solution on the neighborhoods is returned to the base grid in a conservative fashion. The advantage of this approach is that it uses only basic mesh information that is already available in many cut cell codes and does not require complex geometric manipulations. Finally, we present a number of test problems that demonstrate the encouraging potential of this technique for applications on curvilinear embedded geometries. Numerical experiments reveal that our scheme converges with order $p+1$ in $L_1$ and between $p$ and $p+1$ in $L_\infty$ for problems with smooth solutions. We also demonstrate that state redistribution is capable of capturing shocks.