Performance Enhancement for High-Order Gas-Kinetic Scheme Based on WENO-Adaptive-Order Reconstruction (Communications in Computational Physics, 2020)
High-order gas-kinetic scheme (HGKS) has been well-developed in the pastyears. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness.However, there is still room for its further improvement. Firstly, the reconstructionin the previous scheme mainly achieves a fifth-order accuracy for the point-wise values at a cell interface due to the use of standard WENO reconstruction, and the slopesof the initial non-equilibrium states have to be reconstructed from the cell interfacevalues and cell averages again. The same order of accuracy for slopes as the originalWENO scheme cannot be achieved. At the same time, the equilibrium state in spaceand time in HGKS has to be reconstructed separately. Secondly, it is complicated to getreconstructed 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 requiresthe slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previousHGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO withadaptive order (WENO-AO) [2] method is implemented for reconstruction. Equippedwith WENO-AO reconstruction, the performance enhancement of HGKS is fully explored. WENO-AO not only provides the interface values, but also the slopes. In otherwords, a whole polynomial inside each cell is provided by the WENO-AO reconstruction. The available polynomial may not benefit to the high-order schemes based on theRiemann solver, where only points-wise values at the cell interface are needed. But,it can be fully utilized in the HGKS. As a result, the HGKS becomes simpler than theprevious one with the direct implementation of cell interface values and their slopesfrom WENO-AO. The additional reconstruction of equilibrium state at the beginningof each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes theabove existing problems in the previous HGKS. The accuracy of the scheme from 1Dto 3D from the new HGKS can recover the theoretical order of accuracy of the WENOreconstruction. In the two- and three-dimensional simulations, the new HGKS showsbetter robustness and efficiency than the previous scheme in all test cases.
Keywords: High-order finite volume scheme; gas-kinetic scheme; WENO reconstruction; high-order Navier-Stokes solver.