High-order Hadamard-form entropy stable multidimensional summation-by-parts discretizations of the Euler and 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 artificial 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 Navier-Stokes equations. In the absence of heat fluxes, the entropy-split scheme is entropy stable for the 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.

翻译：高精度Hadamard形式熵稳定多维求和分部离散格式在求解欧拉方程与纳维-斯托克斯方程时，其计算成本显著高于标准散度形式离散格式。为寻求更高效的熵稳定格式，我们将熵分裂方法推广至非结构化网格并探究其性质。该格式的核心要素包括：Harten熵函数、具有对角范数矩阵的对角$-\mathsf{E}$求和分部算子，以及熵守恒同步逼近项。我们证明，该格式在周期曲线非结构化网格上求解欧拉方程时具有高精度与熵守恒特性。通过构建熵稳定矩阵型人工耗散算子，可将其添加至同步逼近项以得到熵稳定半离散格式。采用松弛龙格-库塔方法实现全离散熵守恒。针对纳维-斯托克斯方程，我们发展了适用于Hadamard形式与熵分裂格式的熵稳定黏性同步逼近项。在无热通量条件下，熵分裂格式对纳维-斯托克斯方程具有熵稳定性。通过熵稳定混合格式确保间断区域附近的局部守恒性。通过涵盖光滑解与间断解的多个数值算例验证理论结果。计算成本对比研究表明，相较于Hadamard形式多维SBP-SAT离散格式，熵分裂格式具有显著的效率优势。