Hybridizing WENO implementations of interpolation and reconstruction-wise operation for upwind-biased schemes with free-stream preservation

Abstract

Cases have shown that WENO schemes usually behave robustly on problems containing shocks with high pressure ratios when uniformed or smooth grids are present, while nonlinear schemes based on WENO interpolations might relatively be liable to numerical instability. In the meanwhile, the latter have manifested their advantages in computations on grids of bad quality, because the free-stream preservation is easily realized there, and what is more flux-splitting schemes with low dissipations can be engaged inherently as well. Targeting at above dissatisfactions, a method by hybridizing WENO implementations of interpolation and reconstruction-wise operation for upwind-biased schemes with flux splitting employed is proposed and corresponding third-, fifth- and seventh-order upwind-biased schemes are proposed. Based on the understandings of [Q. Li, et al. Commun. Comput. Phys. 22 (2017) 64-94], the free-stream preservation of proposed schemes is achieved with incorporation of frozen grid metrics in WENO reconstructions-wise operations on split fluxes. In proposed schemes, flux-splitting schemes with low dissipation can also be applied for the flux on a cell edge. As a byproduct, an implementation of WENO scheme with free-stream preservation is obtained. Numerical examples are provided as following with the third- and fifth-order schemes being tested. In tests of free-stream preservation, the property is achieved as expected (including two implementations of WENO). The computation of 1-D Sod problem shows the capability of proposed schemes on solving ordinary shock discontinuity. 2-D vortex preservation and double Mach reflection are tested on uniformed and randomized grids. The accomplishment by proposed schemes manifests their capability and robustness on solving problems under rigorous circumstances.