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We present an extension of the summation-by-parts (SBP) framework totensor-product spectral-element operators in collapsed coordinates. Theproposed approach enables the construction of provably stable discretizationsof arbitrary order which combine the geometric flexibility of unstructuredtriangular and tetrahedral meshes with the efficiency of sum-factorizationalgorithms. Specifically, a methodology is developed for constructingtriangular and tetrahedral spectral-element operators of any order whichpossess the SBP property (i.e. satisfying a discrete analogue of integration byparts) as well as a tensor-product decomposition. Such operators are thenemployed within the context of discontinuous spectral-element methods based onnodal expansions collocated at the tensor-product quadrature nodes as well asmodal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latterapproach resolving the time step limitation associated with the singularity ofthe collapsed coordinate transformation. Energy-stable formulations forcurvilinear meshes are obtained using a skew-symmetric splitting of the metricterms, and a weight-adjusted approximation is used to efficiently invert thecurvilinear modal mass matrix. The proposed schemes are compared to those usingnon-tensorial multidimensional SBP operators, and are found to offer comparableaccuracy to such schemes in the context of smooth linear advection problems oncurved meshes, but at a reduced computational cost for higher polynomialdegrees.

Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra