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 highorder SBP discretizations to multiple dimensions has been limited to tensor product domains. This paper presents a definition for multidimensional 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 multidimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences between these SBP operators and those typically used in spectral-element, spectral-difference, and nodal discontinuous Galerkin methods are discussed. An assembly process is described that builds 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.