A unifying algebraic framework for discontinuous Galerkin and flux reconstruction methods based on the summation-by-parts property
Tristan Montoya, David W. Zingg
TL;DR
This work formulates a unifying matrix-based SBP framework for DG and FR methods applied to conservation laws on unstructured grids, clarifying how multidimensional SBP operators ensure discrete equivalence between strong and weak forms and govern conservation and energy stability. It extends known strong-weak form equivalence beyond collocation DG with LG/LGL quadrature to general nodal/modal DG and the VCJH family of FR schemes, and provides new algebraic proofs of conservation and energy stability under suitable quadrature and norm conditions. The framework presents a generalized algebraic formulation encompassing DG and VCJH FR via lifting matrices and projection operators, establishing a bridge between strong-form FR and filtered DG schemes. Numerical demonstrations on two-dimensional linear advection and Euler equations corroborate the theoretical results and illustrate design choices within the unified approach. The results offer a principled basis for cross-method analysis, potential efficiency gains, and avenues toward convergence theory and extensions to nonlinear, diffusive, or entropy-stable settings.
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.