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Summation-by-parts (SBP) finite-difference discretizations share manyattractive properties with Galerkin finite-element methods (FEMs), includingtime stability and superconvergent functionals; however, unlike FEMs, SBPoperators are not completely determined by a basis, so the potential exists totailor SBP operators to meet different objectives. To date, application ofhigh-order SBP discretizations to multiple dimensions has been limited totensor product domains. This paper presents a definition for multi-dimensionalSBP finite-difference operators that is a natural extension of one-dimensionalSBP operators. Theoretical implications of the definition are investigated forthe special case of a diagonal norm (mass) matrix. In particular, adiagonal-norm SBP operator exists on a given domain if and only if there is acubature rule with positive weights on that domain and the polynomial-basismatrix has full rank when evaluated at the cubature nodes. Appropriatesimultaneous-approximation terms are developed to impose boundary conditionsweakly, and the resulting discretizations are shown to be time stable. Concreteexamples of multi-dimensional SBP operators are constructed for the triangleand tetrahedron; similarities and differences with spectral-element andspectral-difference methods are discussed. An assembly process is describedthat builds diagonal-norm SBP operators on a global domain from element-leveloperators. Numerical results of linear advection on a doubly periodic domaindemonstrate the accuracy and time stability of the simplex operators.

Multidimensional Summation-By-Parts Operators: General Theory and Application to Simplex Elements