  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.
