A unifying algebraic framework for discontinuous Galerkin and flux reconstruction methods based on the summation-by-parts property

TL;DR

This paper introduces a unifying algebraic framework based on the summation-by-parts property for analyzing and establishing conservation and stability of discontinuous Galerkin and flux reconstruction methods on unstructured grids.

Contribution

It extends the algebraic analysis of DG and FR methods using SBP properties, unifying their formulations and stability proofs under a common framework.

Findings

Demonstrates the equivalence of strong and weak formulations for various DG and FR schemes.

Provides new algebraic proofs of conservation and energy stability for these methods.

Validates the theoretical results with numerical experiments on 2D linear advection and Euler equations.

Abstract

We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136-155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented, in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.