  We present an extension of the summation-by-parts (SBP) framework to
tensor-product spectral-element operators in collapsed coordinates. The
proposed approach enables the construction of provably stable discretizations
of arbitrary order which combine the geometric flexibility of unstructured
triangular and tetrahedral meshes with the efficiency of sum-factorization
algorithms. Specifically, a methodology is developed for constructing
triangular and tetrahedral spectral-element operators of any order which
possess the SBP property (i.e. satisfying a discrete analogue of integration by
parts) as well as a tensor-product decomposition. Such operators are then
employed within the context of discontinuous spectral-element methods based on
nodal expansions collocated at the tensor-product quadrature nodes as well as
modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter
approach resolving the time step limitation associated with the singularity of
the collapsed coordinate transformation. Energy-stable formulations for
curvilinear meshes are obtained using a skew-symmetric splitting of the metric
terms, and a weight-adjusted approximation is used to efficiently invert the
curvilinear modal mass matrix. The proposed schemes are compared to those using
non-tensorial multidimensional SBP operators, and are found to offer comparable
accuracy to such schemes in the context of smooth linear advection problems on
curved meshes, but at a reduced computational cost for higher polynomial
degrees.
