WENO schemes based on upwind and centred TVD fluxes

In this paper we propose to use second-order TVD fluxes, instead of first-order monotone fluxes, in the framework of finite-volume weighted essentially non-oscillatory (WENO) schemes. We call the new improved schemes the WENO-TVD schemes. They include both upwind and centred schemes on non-staggered meshes. Numerical results suggest that our schemes are superior to the original schemes used with first-order monotone fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.