  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.
