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

Abstract

A generalized algebraic framework is presented for a broad class of high-order methods for hyperbolic systems of conservation laws on curvilinear unstructured grids. The framework enables the unified analysis of many popular discontinuous Galerkin (DG) and flux reconstruction (FR) schemes based on properties of the matrix operators constituting such discretizations. The salient components of the proposed methodology include the formulation of a polynomial approximation space and its representation through a nodal or modal basis on the reference element, the construction of discrete inner products and projection operators based on quadrature or collocation, and the weak enforcement of boundary and interface conditions using numerical flux functions. Situating such components common to DG and FR methods within the context of the summation-by-parts property, a discrete analogue of integration by parts, we establish the algebraic equivalence of certain strongform and weak-form discretizations, reinterpret and generalize existing connections between the DG and FR methods, and describe a unifying approach for the analysis of conservation and linear stability for methods within the present framework. Numerical experiments are presented for the two-dimensional linear advection and compressible Euler equations, corroborating the theoretical results.

Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

“…This differs from the literature where the ESFR correction functions were only used to reconstruct the flux on the surface [1,2,50,57,58,3,5,59,53]. In addition, the proposed scheme is in contrast from schemes where the ESFR norm 6 was applied to the conservative discretization; either filtering the strong form surface integral [60,47,20,52], or filtering the entire weak form [22]; since such stated schemes are only linearly stable.…”

“…SBP operators are matrix difference operators that are mimetic to high-order integration by parts and when combined with appropriate interface coupling procedures, for example simultaneous approximation terms (SATs) [12,13,14,15,16,17,18,19], lead to provably stable and conservative methods. FR has been cast in SBP form [20,21,22] as well in residual distribution schemes [23,24,25,26] paving the way for a common framework to analyze high-order schemes. Moreover, discretizations having the SBP property form the foundations for nonlinearly stable schemes for nonlinear conservation laws [27,28,29,30,18,19,31,32,33,34,35,36,37,38].…”