Summation-by-parts (SBP) operators have a number of properties that make them an attractive option for higher-order spatial discretizations of partial differential equations. In particular, they enable the derivation of higher-order boundary closures leading to provable time stability. When implemented on multi-block structured meshes in conjunction with simultaneous approximation terms (SATs)-penalty terms that impose boundary and interblock-coupling conditions in a weak sense-they offer additional properties of value, even for second-order accurate schemes and steady problems. For example, they involve low communication overhead for efficient parallel algorithms and relax the continuity requirements of both the mesh and the solution across block interfaces. This paper provides a brief history of seminal contributions to, and applications of, SBP-SAT methods followed by a description of their properties and a methodology for deriving SBP operators for first derivatives and second derivatives with variable coefficients. A procedure for deriving SATs is also provided. Practical aspects are discussed, including artificial dissipation, transformation to curvilinear coordinates, and application to the Navier-Stokes equations. Recent developments are reviewed, including a variational interpretation, the connection to quadrature rules, functional superconvergence, error estimates, and dual consistency. Finally, the connection to quadrature rules is exploited to provide a generalization of the SBP concept to a broader class of operators, enabling a unification and rigorous development of SATs for operators such as nodal-based pseudo-spectral and some discontinuous Galerkin operators.

Abstract. Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely determined by a basis, so the potential exists to tailor SBP operators to meet different objectives. To date, application of highorder SBP discretizations to multiple dimensions has been limited to tensor product domains. This paper presents a definition for multi-dimensional SBP finite-difference operators that is a natural extension of one-dimensional SBP operators. Theoretical implications of the definition are investigated for the special case of a diagonal norm (mass) matrix. In particular, a diagonal-norm SBP operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial-basis matrix has full rank when evaluated at the cubature nodes. Appropriate simultaneous-approximation terms are developed to impose boundary conditions weakly, and the resulting discretizations are shown to be time stable. Concrete examples of multi-dimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences with spectralelement and spectral-difference methods are discussed. An assembly process is described that builds diagonal-norm SBP operators on a global domain from element-level operators. Numerical results of linear advection on a doubly periodic domain demonstrate the accuracy and time stability of the simplex operators.Key words. summation-by-parts, finite-difference method, unstructured grid, spectral-element method, spectral-difference method, mimetic discretization AMS subject classifications. 65N06, 65M60, 65N121. Introduction. Summation-by-parts (SBP) operators are high-order finitedifference schemes that mimic the symmetry properties of the differential operators they approximate [19]. Respecting such symmetries has important implications; in particular, they enable SBP discretizations that are both time stable and high-order accurate [4,34,27], properties that are essential for robust, long-time simulations of turbulent flows [25,35].Most existing SBP operators are one-dimensional [30,24,31,23] and are applied to multi-dimensional problems using a multi-block tensor-product formulation [32,14,29]. Like other tensor-product methods, the restriction to multi-block grids complicates mesh generation and adaptation, and it limits the geometric complexity that can be considered in practice.The limitations of the tensor-product formulation motivate our interest in generalizing SBP operators to unstructured grids. There are two ways this generalization has been pursued in the literature: 1) construct global SBP operators on an arbitrary distribution of nodes, or; 2) construct SBP operators on reference elements and assemble a global discretization by coupling these smaller elements.The first approach is appealing conceptually, and it is certainly viable for secondor...

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The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points [1,2,3] and their equivalence with summation-by-parts (SBP) finite difference operators [4]. In this work, we show how to efficiently generalize the construction of semi-discretely entropy stable schemes on tensor product elements to Gauss points and generalized SBP operators. Numerical experiments suggest that the use of Gauss points significantly improves accuracy on curved meshes.