  We present a new class of efficient and robust discontinuous spectral-element
methods of arbitrary order for nonlinear hyperbolic systems of conservation
laws on curved triangular and tetrahedral unstructured grids. Such
discretizations employ a recently introduced family of sparse tensor-product
summation-by-parts (SBP) operators in collapsed coordinates within an
entropy-stable modal formulation. The proposed algorithms exploit the structure
of such SBP operators alongside that of the Proriol-Koornwinder-Dubiner
polynomial basis, and a weight-adjusted approximation is used to efficiently
invert the local mass matrix for curvilinear elements. Using such techniques,
the number of required entropy-conservative two-point flux evaluations between
pairs of quadrature nodes is significantly reduced relative to existing
entropy-stable formulations using (non-tensor-product) multidimensional SBP
operators, particularly for high polynomial degrees, with an improvement in
time complexity from $\mathcal{O}(p^{2d})$ to $\mathcal{O}(p^{d+1})$, where $p$
is the polynomial degree of the approximation and $d$ is the number of spatial
dimensions. In numerical experiments involving smooth solutions to the
compressible Euler equations, the proposed tensor-product schemes demonstrate
similar levels of accuracy for a given mesh and polynomial degree to those
using multidimensional SBP operators based on symmetric quadrature rules.
Furthermore, both operator families are shown to give rise to entropy-stable
methods which exhibit excellent robustness for under-resolved problems. Such
results suggest that the algorithmic advantages resulting from the use of
tensor-product operators are obtained without compromising accuracy or
robustness, enabling the efficient extension of the benefits of entropy
stability to higher polynomial degrees than previously considered for
triangular and tetrahedral elements.
