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.

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-...

In this paper, a third-order weighted essentially non-oscillatory (WENO) scheme is developed for hyperbolic conservation laws on unstructured quadrilateral and triangular meshes. As a starting point, a general stencil is selected for the cell with any local topology, and a unified linear scheme can be constructed. However, in the traditional WENO scheme on unstructured meshes, the very large and negative weights may appear for the mesh with lower quality, and the very large weights make the WENO scheme unstable even for the smooth tests. In the current scheme, an optimization approach is given to deal with the very large linear weights, and the splitting technique is considered to deal with the negative weights obtained by the optimization approach. The non-linear weight with a new smooth indicator is proposed as well, in which the local mesh quality and discontinuities of solutions are taken into account simultaneously. Numerical tests are presented to validate the current scheme. The expected convergence rate of accuracy is obtained, and the absolute value of error is not affected by mesh quality. The numerical tests with strong discontinuities validate the robustness of current WENO scheme.