In this paper, the entropy conservative/stable algorithms presented by Del Rey Fernandez and coauthors [18,16,17] for the compressible Euler and Navier-Stokes equations on nonconforming p-refined/coarsened curvilinear grids is extended to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, a computationally simple and efficient approach based upon using decoupled interpolation operators is utilized. The resulting scheme is entropy conservative/stable and element-wise conservative. Numerical simulations of the isentropic vortex and viscous shock propagation confirm the entropy conservation/stability and accuracy properties of the method (achieving ~ p + 1 convergence) which are comparable to those of the original conforming scheme [4,35]. Simulations of the Taylor-Green vortex at Re = 1,600 and turbulent flow past a sphere at Re = 2,000 show the robustness and stability properties of the overall spatial discretization for unstructured grids. Finally, to demonstrate the entropy conservation property of a fully-discrete explicit entropy stable algorithm with h/p refinement/coarsening, we present the time evolution of the entropy function obtained by simulating the propagation of the isentropic vortex using a relaxation Runge-Kutta scheme.

中文翻译：

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可压缩 Navier-Stokes 方程的具有分部求和性质的熵稳定 p-非一致性离散化

在本文中，由 Del Rey Fernandez 和合著者 [18,16,17] 提出的熵保守/稳定算法，用于可压缩 Euler 和 Navier-Stokes 方程在非一致 p​​ 细化/粗化曲线网格上的扩展到 h/p 细化/粗化。开发不合格算法的主要困难是跨不合格接口构建适当的耦合程序。这里，利用了基于使用解耦插值算子的计算上简单且有效的方法。得到的方案是熵保守/稳定和元素保守的。等熵涡旋和粘性冲击传播的数值模拟证实了该方法的熵守恒/稳定性和精度特性（实现 ~ p + 1 收敛），它们与原始符合方案 [4,35] 的那些特性相当。对 Re = 1,600 处的 Taylor-Green 涡旋和 Re = 2,000 处通过球体的湍流的模拟显示了非结构化网格的整体空间离散化的稳健性和稳定性特性。最后，为了证明具有 h/p 细化/粗化的完全离散显式熵稳定算法的熵守恒性质，我们展示了通过使用松弛 Runge-Kutta 方案模拟等熵涡旋的传播而获得的熵函数的时间演化. 600 和经过 Re = 2,000 球体的湍流显示了非结构化网格的整体空间离散化的稳健性和稳定性特性。最后，为了证明具有 h/p 细化/粗化的完全离散显式熵稳定算法的熵守恒性质，我们展示了通过使用松弛 Runge-Kutta 方案模拟等熵涡旋的传播而获得的熵函数的时间演化. 600 和经过 Re = 2,000 球体的湍流显示了非结构化网格的整体空间离散化的稳健性和稳定性特性。最后，为了证明具有 h/p 细化/粗化的完全离散显式熵稳定算法的熵守恒性质，我们展示了通过使用松弛 Runge-Kutta 方案模拟等熵涡旋的传播而获得的熵函数的时间演化.