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

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 strong-form 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.