  High-order Hadamard-form entropy stable multidimensional summation-by-parts
discretizations of the Euler and compressible Navier-Stokes equations are
considerably more expensive than the standard divergence-form discretization.
In search of a more efficient entropy stable scheme, we extend the
entropy-split method for implementation on unstructured grids and investigate
its properties. The main ingredients of the scheme are Harten's entropy
functions, diagonal-$ \mathsf{E} $ summation-by-parts operators with diagonal
norm matrix, and entropy conservative simultaneous approximation terms (SATs).
We show that the scheme is high-order accurate and entropy conservative on
periodic curvilinear unstructured grids for the Euler equations. An entropy
stable matrix-type interface dissipation operator is constructed, which can be
added to the SATs to obtain an entropy stable semi-discretization.
Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta
method. Entropy stable viscous SATs, applicable to both the Hadamard-form and
entropy-split schemes, are developed for the compressible Navier-Stokes
equations. In the absence of heat fluxes, the entropy-split scheme is entropy
stable for the compressible Navier-Stokes equations. Local conservation in the
vicinity of discontinuities is enforced using an entropy stable hybrid scheme.
Several numerical problems involving both smooth and discontinuous solutions
are investigated to support the theoretical results. Computational cost
comparison studies suggest that the entropy-split scheme offers substantial
efficiency benefits relative to Hadamard-form multidimensional SBP-SAT
discretizations.
