Performance Enhancement for High-order Gas-kinetic Scheme Based on WENO-adaptive-order Reconstruction

High-order gas-kinetic scheme (HGKS) has been well-developed in the past years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness. However, there are still rooms for its further improvement. Firstly, the reconstruction in the previous scheme mainly achieves a third-order accuracy for the initial non-equilibrium states. At the same time, the equilibrium state in space and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with adaptive order (WENO-AO) method is implemented for reconstruction.A whole polynomial inside each cell is provided in WENO-AO reconstruction. The HGKS becomes simpler than the previous one with the direct implementation of cell interface values and their slopes from WENO-AO. The additional reconstruction of equilibrium state at the beginning of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the above existing problems in previous HGKS. The accuracy of the scheme from 1D to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO reconstruction.In the two- and three-dimensional simulations, the new HGKS shows better robustness and efficiency than the previous scheme in all test cases.