A theoretical analysis of the entropy conservation properties is conducted to explain the different behaviors of the non-dissipative finite-difference spatial discretization schemes, such as the kinetic-energy and entropy preserving (KEEP) schemes. The analysis is conducted based on the spatially-discretized entropy-evolution equation derived from the Euler equations with retaining the discrete-level strictness. The present analysis shows that the analytical relations (ARs) employed by the KEEP schemes eliminate some terms of the discretized entropy-evolution equation and help simplify the equation, while the ARs are not sufficient to explain the entropy-conservation property. Therefore, the entropy error is decomposed into several terms, and the behaviors of those decomposed error terms are observed in the compressible inviscid Taylor–Green vortex test case. The results of the test case show that the terms containing the velocity difference between two grid points cause significant entropy conservation error and result in the different entropy conservation properties of the tested non-dissipative schemes. Furthermore, the KEEP schemes are redefined based on the present entropy-error analysis. In the redefined KEEP schemes, the formulation does not contain logarithmic means, and the strictness of entropy conservation can be adjusted easily by the truncation order of the Maclaurin expansions. Finally, the present entropy-error analysis and the redefined KEEP schemes are extended to the generalized curvilinear coordinates.

对熵守恒性质进行了理论分析，以解释非耗散有限差分空间离散方案的不同行为，例如动能和熵保持（KEEP）方案。分析是基于从欧拉方程导出的空间离散熵演化方程进行的，并保留了离散级的严格性。目前的分析表明，KEEP方案采用的分析关系（ARs）消除了离散熵演化方程的一些项，有助于简化方程，而ARs不足以解释熵守恒性质。因此，熵误差被分解为几项，并且在可压缩的非粘性泰勒-格林涡旋测试用例中观察到这些分解的误差项的行为。测试案例的结果表明，包含两个网格点之间速度差异的项会导致显着的熵守恒误差，并导致所测试的非耗散方案的熵守恒性质不同。此外，基于目前的熵误差分析重新定义了 KEEP 方案。在重新定义的 KEEP 方案中，公式不包含对数均值，熵守恒的严格性可以通过 Maclaurin 展开式的截断顺序轻松调整。最后，将当前的熵误差分析和重新定义的 KEEP 方案扩展到广义曲线坐标。测试案例的结果表明，包含两个网格点之间速度差异的项会导致显着的熵守恒误差，并导致所测试的非耗散方案的熵守恒性质不同。此外，基于目前的熵误差分析重新定义了 KEEP 方案。在重新定义的 KEEP 方案中，公式不包含对数均值，熵守恒的严格性可以通过 Maclaurin 展开式的截断顺序轻松调整。最后，将当前的熵误差分析和重新定义的 KEEP 方案扩展到广义曲线坐标。测试案例的结果表明，包含两个网格点之间速度差异的项会导致显着的熵守恒误差，并导致所测试的非耗散方案的熵守恒性质不同。此外，基于目前的熵误差分析重新定义了 KEEP 方案。在重新定义的 KEEP 方案中，公式不包含对数均值，熵守恒的严格性可以通过 Maclaurin 展开式的截断顺序轻松调整。最后，将当前的熵误差分析和重新定义的 KEEP 方案扩展到广义曲线坐标。