In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral-like resolution will be presented. Based on the high-order gas evolution model, both the flux function and conservative flow variables in GKS can be evaluated explicitly from the time-accurate gas distribution function at a cell interface. As a result, inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step. The flow variable update and slope update are coming from the same physical solution at the cell interface. This strategy needs time accurate solution at a cell interface, which cannot be achieved by the Riemann problem based flow solvers, even though they can also provide the interface flux functions and interface flow variables. Instead, in order to update the slopes in the Riemann-solver based schemes, such as HWENO, there are additional governing equations for slopes or equivalent degrees of freedom inside each cell. In GKS, only a single time accurate gas evolution model is needed at the cell interface for updating cell averaged flow variables through interface fluxes and updating the cell averaged slopes through the interface flow variables. Based on both cell averaged values and their slopes, compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed. As analyzed in this paper, the local linear compact reconstruction without limiter can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives. For nonlinear gas dynamic evolution, in order to avoid spurious oscillation in discontinuous region, the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction. Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction. In GKS, the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation, which covers a physical process from an initial non-equilibrium state to a final equilibrium one. Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction, and the equilibrium state is basically constructed from the linear symmetric reconstruction, the GKS evolution models unifies the nonlinear and linear reconstructions in a gas relaxation process in the determination of a time-dependent gas distribution function. This property gives GKS great advantages in capturing both discontinuous shock waves and the linear aero-acoustic waves in a single computation due to its dynamical adaptation of non-equilibrium and equilibrium states in different flow regions. This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions. Compared with discontinuous Galerkin (DG) scheme, the current compact GKS uses the same local and compact stencil, achieves the 6th-order and 8th-order accuracy, uses a much larger time step with CFL number ≥ 0.3, has the robustness as a 2nd-order scheme, and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process. The nonlinear reconstruction in the compact GKS is solely based on the WENO-Z technique. At the same time, the current scheme solves the Navier-Stokes equations automatically due to combined inviscid and viscous flux terms from a single time evolution gas distribution function at a cell interface. Due to the use of multi-stage multi-derivative (MSMD) time-stepping technique, for achieving a 4th-order time accuracy, the current scheme uses only two stages instead of four in the traditional Runge-Kutta method. As a result, the current GKS becomes much more efficient than the corresponding same order DG method. A variety of numerical tests are presented to validate the compact 6th and 8th-order GKS. The current scheme presents a state-of-art numerical solutions under a wide range of flow conditions, i.e., strong shock discontinuity, shear instability, aero-acoustic wave propagation, and NS solutions. It promotes the development of high-order scheme to a new level of maturity. The success of the current scheme crucially depends on the high-order gas evolution model, which cannot be achieved by any other approach once the 1st-order Riemann flux function is still used in the development of high-order numerical algorithms.

中文翻译：

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紧凑的高阶气体动力学方案，具有类似光谱的分辨率，用于可压缩流模拟

在本文中，将提出一类具有类似光谱分辨率的紧凑型高阶气体动力学方案（GKS）。基于高阶气体逸出模型，可以从单元界面处的时间精确气体分布函数中明确评估GKS中的通量函数和保守流量变量。结果，在每个控制容积内，可以在每个时间步长内更新单元平均流量变量及其单元平均梯度。流量变量更新和斜率更新来自单元接口处的相同物理解决方案。此策略需要在单元界面处具有时间精确的解决方案，即使基于Riemann问题的流求解器也可以提供界面通量函数和界面流变量，但该解决方案无法实现。代替，为了更新基于Riemann-solver的方案（例如HWENO）中的斜率，每个像元内存在其他斜率或等效自由度的控制方程。在GKS中，仅需要一个时间精确的气体逸出模型在单元界面上，即可通过界面通量更新单元平均流量变量，并通过界面流量变量更新单元平均斜率。基于单元均值及其斜率，可以开发紧凑的6阶和8阶线性和非线性重构。正如本文所分析的，与没有建立具有全局耦合流量变量及其导数的Lele紧凑型方案相比，没有限制器的局部线性紧凑型重构可以在大波数下实现类似频谱的分辨率。对于非线性气体动力学演化，为了避免在不连续区域中的寄生振荡，可以将上述从对称模板进行的紧凑线性重构分为多个子模板，并应用有偏的非线性WENO-Z重构。因此，可以通过6阶和8阶紧凑型WENO型非线性重构来捕获不连续解。在GKS中，单元界面处气体分布函数的时间演化解基于动力学模型方程的积分解，该模型涵盖了从初始非平衡态到最终平衡态的物理过程。由于基于非线性WENO-Z重建获得了初始非平衡状态，并且基本上由线性对称重建构成了平衡状态，在确定随时间变化的气体分布函数时，GKS演化模型统一了气体弛豫过程中的非线性和线性重构。由于其动态适应不同流动区域中的非平衡态和平衡态，该特性为GKS在一次计算中捕获不连续冲击波和线性空气声波方面提供了巨大优势。这种动态自适应模型有助于解决有关线性和非线性重构选择的高阶方案开发中的一个长期问题。与不连续的Galerkin（DG）方案相比，当前的紧凑型GKS使用相同的局部和紧凑型模板，达到6阶和8阶精度，使用更大的时间步长（CFL数≥0.3），并且具有鲁棒性。订单方案 并在冲击区域和平滑区域获得准确的解决方案，而无需引入故障单元和其他限制流程。紧凑型GKS中的非线性重建完全基于WENO-Z技术。同时，由于来自单元界面单次逸出气体分布函数的无粘性和粘性通量项的组合，当前方案自动求解Navier-Stokes方程。由于使用了多级多导数（MSMD）时间步进技术，为了实现四阶时间精度，当前方案仅使用两个阶段，而不是传统的Runge-Kutta方法中的四个阶段。结果，当前的GKS比相应的相同DG方法效率更高。提出了各种数值测试来验证紧凑的6阶和8阶GKS。当前方案提供了在广泛流动条件下的最新数值解，即强冲击不连续性，剪切不稳定性，航空声波传播和NS解。它促进了高阶计划的发展到新的成熟水平。当前方案的成功关键取决于高阶气体释放模型，一旦在高阶数值算法的开发中仍使用一阶黎曼通量函数，则该方法无法通过任何其他方法来实现。