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

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 high-order 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 spectral-element 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.