An Alternative Analysis of Discontinuous Galerkin Method for Hyperbolic Conservation Law

TL;DR

This paper offers an alternative analysis of the Discontinuous Galerkin method for hyperbolic conservation laws, aiming to better understand its physical mechanisms and address stability issues in CFD applications.

Contribution

It presents a novel analytical approach to the DG method based on the linear advection equation, providing insights into its stability and physical interpretation.

Findings

Enhanced understanding of DG stability in non-smooth regions

Insights into the physical mechanisms of DG updates

Potential improvements for DG method development

Abstract

The development and application of the Discontinuous Galerkin (DG) method have attracted great attention in computational fluid dynamics (CFD) com- munity in the past decades. The underlying reason for such an intensive investigation is due to favorable properties of the DG method, such as higher order, compactness, and easy parallelization. However, for the compressible flow simulations, the DG method is also associated with unfavorable prop- erties, such as the frequent instabilities in non-smooth regions. Due to the finite element formulation, except the update of the cell averaged conserva- tive flow variable, it becomes rather difficulty to fully understand the physical mechanism in the DG method for the updates of other degrees of freedom in- side each element. In this short note, based on the linear advection equation we are going to analyze the DG method in an alternative way. The results obtained may be useful for the CFD community for its further development of the DG methods.