On the use of TVD fluxes in ENO and WENO schemes

Very high order methods, such as ENO/WENO methods [21, 30, 19], Runge-Kutta Discontinuous Galerkin Finite Element Methods [12] and ADER methods [54,46] often use high order (e.g. fifth order) polynomial reconstruction of the solution of a lower (first) order monotone flux as the building block. In this paper we propose to use second order TVD fluxes in the framework of such methods and apply the principle to the finite-volume ENO, WENO and MPWENO schemes. We call the new improved schemes the ENO-TVD, WENO-TVD and MPWENO-TVD schemes respectively. They include both upwind and centred schemes with non-stuggered meshes. Numerical results suggest that our schemes are superior to original schemes with first order fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.