  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.
