A family of high-order gas-kinetic schemes and its comparison with Riemann solver based high-order methods

Abstract

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.

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

“…The periodic boundary condition is applied in the y direction. The extracted density profile along the horizontal line and the "exact" solution, which is obtained by the 1-D fifth-order WENO GKS [10] with 10,000 grid points at t = 3.8 are shown in Fig.8. Again the current third-order results are as good as the ones from the traditional GKS with the WENO5-Z reconstruction and the same mesh size.…”

“…Recently, with the incorporation of multi-derivative multi-stage technique, a family of HGKS has been developed [10]. Based on the 5th-order WENO reconstruction [11,2], the performance of HGKS shows great advantages in terms of efficiency, accuracy, and robustness in comparison with traditional high-order scheme with Riemann solver, especially in the capturing of shear instabilities due to its multi-dimensionality in the GKS flux function.…”

An efficient high-order gas-kinetic scheme (I): Euler equations

“…It even occupies most of the computation cost in the fourth-order HGKS [14] to calculate the third-order derivatives of particle distribution functions and take their moments. Consequently, the state-of-art HGKS with higher than third-order accuracy is compromised to the multi-stage framework where only low-order HGKS evolution model is utilized [18,9,8].…”

“…The third and fourth-order HGKS have been developed adherently [13,16,14]. However, because of the time-consuming computation process of the sophisticated extension, the state-of-art fourth and fifth-order HGKS is compromised to the multi-stage framework in practice [18,9].…”

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

“…Due to the use of only two reconstructions in the 4thorder scheme, the GKS becomes very efficient in comparison with schemes based on the Runge-Kutta time stepping technique. By combining the second-order or third-order GKS fluxes and MSMD technique again, a family of higher-order gas-kinetic methods has been constructed [22,21].…”