WENO interpolation-based and upwind-biased schemes with free-stream preservation

Abstract

Based on the understandings regarding linear upwind schemes with flux splitting to achieve free-stream preservation (Q. Li, etc. Commun. Comput. Phys., 22 (2017) 64-94), a series of WENO interpolation-based and upwind-biased nonlinear schemes are proposed in this study. By means of engagement of fluxes on midpoints, the nonlinearity of schemes is introduced through WENO interpolations, and upwind-biased features are acquired through the choice of dependent grid stencil. Regarding the third- and fifth-order versions, schemes with one and two midpoints are devised and carefully tested. With the integration of the piecewise-polynomial mapping function methods (Q. Li, etc. Commun. Comput. Phys. 18 (2015) 1417-1444), the proposed schemes are found to achieve the designed orders and free-stream preservation property. In 1-D Sod and Shu-Osher problems, all schemes succeed in yielding well predictions. In 2-D cases, the vortex preservation, supersonic inviscid flow around cylinder at M=4, Riemann problem and Shock-vortex interaction problems are tested. In each problem, two types of grids are employed, i.e. the uniformed/smooth grids and the randomized/partially-randomized grids. On the latter, the shock wave and complex flow structures are located/partially located. All schemes fulfill computations in uniformed/smooth grids with satisfactory results. On randomized grids, all schemes accomplish computations and yield reasonable results except the third-order one with two midpoints engaged fails in Riemann problem and shock-vortex interaction problem. Overall speaking, the proposed schemes manifest the capability to solve problems on grids with bad quality, and therefore indicate their potential in engineering applications.