The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flow in Cartesian mesh. To study the flow problem in general geometry, such as the flow over a wing-body configuration, the development of a three-dimensional HGKS in general curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates. Based on the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and is transformed back to the physical domain for the update of conservative flow variables inside each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables, and the conservative variables are obtained by ratio of the above reconstructed data at Gaussian quadrature points of each cell interface. In the two-stage fourth-order gas kinetic scheme (GKS), similar to the generalized Riemann solver (GRP), the initial spatial derivatives of conservative variables have to be used in the evaluation of the time dependent flux function in GKS, which are reconstructed as well through orthogonalization in physical space and chain rule. A variety of numerical examples from the order tests to the solutions with strong discontinuities are presented to validate the accuracy and robustness of the current scheme. The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is also demonstrated through the numerical example.
High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids
Ningbo University（宁波大学）；Northeast Normal University（东北师范大学）；Tsinghua University（清华大学）
Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement
Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier-Stokes Equations
将Del Rey Fernandez et al.（2019）的双曲守恒律的熵守恒、曲线、双曲、p-精化算法从可压缩Euler方程推广到可压缩Navier-斯托克斯方程。采用了一种简单而灵活的耦合方法，相邻单元之间采用平面插值算子。通过最小化过程数值近似了曲率体积度量项，并满足离散几何守恒律条件。不同的曲线表面度量用于相邻的接口，以构建接口耦合项，从而本地化的离散几何守恒律约束到每个单独的元素。所得到的方案是熵保守/稳定的，元素保守的，和自由度保持。粘性界面耗散算子的开发，保持了熵的稳定性的基础计划。在粘性激波问题、雷诺数Re= 1，600时的泰勒-格林涡问题和Re = 2，000时的亚音速湍流流过球体的情况下，所得到的数值格式的精度和稳定性与原来的协调格式（达到~p+1收敛）相当。
Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows
Institute of Mathematics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学数学研究所）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）；Institute for Atmospheric Physics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学大气物理研究所）
Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions
Freie Universität Berlin（柏林自由大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；Technical University of Darmstadt（达姆施塔特工业大学）；Clausthal University of Technology（克劳塔尔工业大学）
Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
介绍了保内积范数松弛龙格-库塔法（David I. Ketcheson，\emph{Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms}，SIAM Journal on数值分析，2019）扩展到一般凸量。通过增加{ \em松弛因子}来加强关于任何凸泛函的守恒、耗散或其他解性质，该松弛参数在每一步乘以龙格库塔更新。此外，保持了其它所需的稳定性（如强稳定性保存）和效率（如低储存要求）性质。该技术可以应用于显式和隐式龙格-库塔法，只需要一个小的修改，现有的实现。在每一步的计算成本是一个额外的标量代数方程的解决方案，一个很好的初始猜测是可用的。该方法的有效性被证明分析和证明在几个数值例子中，包括应用到高阶熵守恒和熵稳定的半离散的非结构网格上的可压缩欧拉和Navier-斯托克斯方程。
Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes equations with variable density and artificial compressibility
Entropy Stable h/p-Nonconforming Discretization with the Summation-by-Parts Property for the Compressible Euler and Navier-Stokes Equations
本文将Del Rey Fernandez等人在曲线网格上求解可压缩Euler和Navier-斯托克斯方程的熵守恒/稳定算法推广到h/p加密/粗化。开发并行计算算法的主要困难是在并行计算接口之间建立适当的耦合过程。在这里，一个计算简单和有效的方法，使用解耦插值算子的基础上利用。所得到的方案是熵保守/稳定和元素保守的。等熵涡和粘性激波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（达到~ p + 1收敛），与原始一致格式[4，35]的性能相当。对Re = 1，600时的泰勒-格林涡和Re = 2，000时通过球体的湍流的模拟显示了非结构网格的整体空间离散的鲁棒性和稳定性。最后，为了证明全离散显式熵稳定算法的熵守恒性质，我们采用松弛Runge-Kutta格式模拟等熵涡的传播，给出了该算法的熵函数随时间的演化.
