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.

翻译：高阶哈达玛形式熵稳定多维求和分部离散方法在求解欧拉与可压缩纳维-斯托克斯方程时，其计算成本显著高于标准散度形式离散方法。为寻求更高效的熵稳定格式，我们将熵分裂方法扩展至非结构化网格并研究其性质。该格式的核心要素包括哈滕熵函数、对角范式矩阵的对角-$\mathsf{E}$ 求和分部算子，以及熵守恒同步逼近项。研究表明，该格式在周期性曲线非结构化网格上对欧拉方程具有高阶精度和熵守恒特性。我们构建了熵稳定的矩阵型界面耗散算子，可通过添加至同步逼近项实现熵稳定半离散化；采用松弛龙格-库塔方法实现完全离散熵守恒。针对可压缩纳维-斯托克斯方程，我们发展了适用于哈达玛形式和熵分裂格式的熵稳定粘性同步逼近项。在无热通量条件下，熵分裂格式对可压缩纳维-斯托克斯方程保持熵稳定性。通过熵稳定混合格式确保间断区域局部守恒性。数值实验涵盖光滑与间断解问题以验证理论结果。计算成本对比研究表明，相比哈达玛形式多维SBP-SAT离散方法，熵分裂格式具有显著效率优势。