Most high order computational fluid dynamics (CFD) methods for compressible flows are based on Riemann solver for the flux evaluation and Runge-Kutta (RK) time stepping technique for temporal accuracy. The main advantage of this kind of approach is the easy implementation and stability enhancement by introducing more middle stages. However, the nth-order time accuracy needs no less than n stages for the RK method, which is very time and memory consuming for a high order method. On the other hand, the multi-stage multi-derivative (MSMD) method can be used to achieve the same order of time accuracy using less middle stages, once the time derivatives of the flux function is used. For the traditional Riemann solver based CFD methods, the lack of time derivatives in the flux function prevents its direct implementation of the MSMD method. However, the gas kinetic scheme (GKS) provides such a time accurate evolution model. By combining the secondorder or third-order GKS flux functions with the MSMD technique, a family of high order gas kinetic methods can be constructed. As an extension of the previous 2-stage 4th-order GKS, the 5th-order schemes with 2 and 3 stages will be developed in this paper. Based on the same 5th-order WENO reconstruction, the performance of gas kinetic schemes from the 2nd-to the 5th-order time accurate methods will be evaluated. The results show that the 5th-order scheme can achieve the theoretical order of accuracy for the Euler equations, and present accurate Navier-Stokes solutions as well due to the coupling of inviscid and viscous terms in the GKS formulation. In comparison with Riemann solver based 5th-order RK method, the high order GKS has advantages in terms of efficiency, accuracy, and robustness, for all test cases. The 4th-order and 5th-order GKS have the same robustness as the 2nd-order scheme for the capturing of discontinuous solutions. The current high order MSMD GKS is a multidimensional scheme with incorporation of both normal and tangential spatial derivatives of flow variables at a cell interface in the flux evaluation. The scheme can be extended straightforwardly to viscous flow computation in unstructured mesh. It provides a promising direction for the development of high-order CFD methods for the computation of complex flows, such as turbulence and acoustics with shock interactions.

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In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier-Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the highorder gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate evolution solution at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the GKS can get the cell averaged slopes inside each control volume as well through the differences of flow variables at the cell interfaces. So, with the updated flow variables and their slopes inside each cell, the HWENO reconstruction can be naturally implemented for the compact high-order reconstruction at the beginning of next step. Therefore, a compact higher-order GKS, such as the two-stages fourth-order compact scheme can be constructed. This scheme is as robust as second-order one, but more accurate solution can be obtained. In comparison with compact fourth-order DG method, the current scheme has only two stages instead of four within each time step for the fourth-order temporal accuracy, and the CFL number used here can be on the order of 0.5 instead of 0.11 for the DG method. Through this research, it concludes that the use of high-order time evolution model rather than the first order Riemann solution is extremely important for the design of robust, accurate, and efficient higher-order schemes for the compressible flows.2 the second-order or third-order GKS fluxes with the multi-stage multi-derivative technique again, a family of high order gas-kinetic methods has been constructed [17]. The above higher-order GKS uses the higher-order WENO reconstruction for spatial accuracy. These schemes are not compact and have room for further improvement.The GKS time dependent gas-distribution function at a cell interface provides not only the flux evaluation and its time derivative, but also time accurate flow variables at a cell interface. The design of compact GKS based on the cell averaged and cell interface values has been conducted before [44,31,32]. In the previous approach, the cell interface values are strictly enforced in the reconstruction, which may not be an appropriate approach. In this paper, inspired by the Hermite WENO (HWENO) reconstruction and compact fourth order GRP scheme [11], instead of using the interface values we are going to get the slopes inside each control volume first, then based on the cell averaged values and slopes inside each control volume the HWENO reconstruction is implemented for the compact highorder reconstruction. The higher-order compact GKS developed in this paper is basically a unified combination of three ingredients, which are the two-stage fourth-order framework for temporal discretization [33], the higher-order gas evolution model for...

In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral resolution will be presented. Based on the high-order gas evolution model in GKS, both the interface flux function and conservative flow variables can be evaluated explicitly from the time-accurate gas distribution function. 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. Different from many other approaches, such as HWENO, there are no additional governing equations in GKS for the slopes or equivalent degrees of freedom independently inside each cell. Therefore, 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 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 gas evolution process for the determination of a time-dependent gas distribution function, which gives great advantages in capturing both discontinuous shock wave and the linear aero-acoustic wave in a single computation due to dynamical adaptation of non-equilibrium and equilibrium state from GKS evolution model in different 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 1 arXiv:1901.00261v1 [physics.comp-ph] 2 Jan 2019 (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 scheme is solely based on the WENO-...

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Following the development of a third-order compact gas-kinetic scheme (GKS) for the Euler and Navier-Stokes equations (Journal of Computational Physics 410 (2020) 109367), in this paper an even higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-accurate flux function, but also the time-evolving flow variables. Therefore, the cell-averaged flow variables and their gradients can be explicitly updated at the next time level from the moments of the same time-dependent gas distribution function. Based on the cell averages and cellaveraged derivatives, both linear and nonlinear high-order reconstruction can be obtained for macroscopic flow variables in the evaluation of local equilibrium and non-equilibrium states. The current nonlinear reconstruction is a combination of WENO and ENO methodology, which is specifically suitable for compact GKS on unstructured mesh with a high-order (≥ 4) accuracy. The initial piecewise discontinuous reconstruction is used for the determination non-equilibrium state and an evolved smooth reconstruction for the equilibrium state. The evolution model in gas-kinetic scheme is based on a relaxation process from non-equilibrium to equilibrium state. The time-accurate gas distribution function in GKS provides the Navier-Stokes flux function directly without separating the inviscid and viscous terms, which simplifies the numerical method on unstructured mesh. Based on the timeaccurate flux solver, the two-stage fourth-order time discretisation can be applied to get a fourth-order time-accurate solution with only two stages, which reduces two reconstructions in comparison with the same time-accurate method with Runge-Kutta time stepping. The current high-order GKS can uniformly capture acoustic and shock waves without identifying trouble cells and implementing additional limiting procedure. In addition, the fourth-up to sixth-order compact GKS can use almost the same time step as a second-order shock capturing scheme. The fourth-order GKS on unstructured mesh will be used in the computations 1 from low speed incompressible viscous flow to the high Mach number shock interaction. The accuracy, efficiency, and robustness of the scheme have been validated. The main conclusion of the paper is that beyond the first-order Riemann solver, the use of high-order gas evolution model seems necessary in the development of high-order schemes.

In this paper, a third-order compact gas-kinetic scheme (GKS) on unstructured tetrahedral mesh is constructed for the compressible Euler and Navier-Stokes solutions. The timedependent gas distribution function at a cell interface is used to calculate the fluxes for the updating the cell-averaged flow variables and to evaluate the time accurate cell-averaged flow variables as well for evolving the cell-averaged gradients of flow variables. With the accurate evolution model for both flow variables and their slopes, the quality of the scheme depends closely on the accuracy and reliability of the initial reconstruction of flow variables. The reconstruction scheme becomes more challenge on tetrahedral mesh, where the conventional second-order unlimited least-square reconstruction can make the scheme be linearly unstable when using cell-averaged conservative variables alone with von Neumann neighbors. Benefiting from the evolved cell-averaged slopes, on tetrahedral mesh the GKS is linearly stable from a compact third-order smooth reconstruction with a large CFL number. In order to further increase the robustness of the high-order compact GKS for capturing discontinuous solution, a new two-step multi-resolution weighted essentially non-oscillatory (WENO) reconstruction will be proposed. The novelty of the reconstruction includes the following. Firstly, it releases the stability issue from a second-order compact reconstruction through the introduction of a pre-reconstruction step. Secondly, in the third-order non-linear reconstruction, only one more large stencil is added beside those in the second-order one, which significantly simplifies the high-order reconstruction. At the same time, the high-order wall boundary treatment is carefully designed by combining the constrained least-square technique and the WENO procedure, where a quadratic element is adopted in the reconstruction for the curved boundary. Numerical tests for both inviscid and viscous flow at low and high speed are presented from the second-order and third-order compact GKS. The proposed third-order scheme shows good robustness in high speed flow computation and favorable mesh adaptability in cases with complex geometry.