Abstract:

The numerical flux determines the performance of numerical methods for solving hyperbolic partial differential equations (PDEs).  In this work, we compare a selection of 8 numerical fluxes in the framework of nonlinear semidiscrete finite volume (FV) schemes, based on Weighted Essentially Non--Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. 
    The methodology is implemented and systematically assessed for order of accuracy in space and time up to seven. The numerical fluxes selected in the present study represent the two existing classes of fluxes, namely centred and upwind. Centred fluxes do not explicitly use wave propagation information, while, upwind fluxes do so from the solution of the Riemann problem via a wave model containing A waves. Upwind fluxes include two subclasses: complete and incomplete fluxes. For complete upwind fluxes,  A = E, where E is the number of characteristic fields in the exact problem.  For incomplete upwind ones,  A < E.    
    Our study is conducted for the one-- and two--dimensional Euler equations, for which we consider the following numerical fluxes: Lax--Friedrichs (LxF), First--Order Centred (FORCE), Rusanov (Rus), Harten--Lax--van Leer (HLL), Central--Upwind (CU), Low--Dissipation Central--Upwind (LDCU), HLLC, and the flux computed through the exact Riemann solver (Ex.RS). We find that the numerical flux has an effect on the performance of the methods. The magnitude of the effect depends on the type of numerical flux and on the order of accuracy of the scheme. It also depends on the type of problem; that is, whether the solution is smooth or discontinuous, whether discontinuities are linear or nonlinear, whether linear discontinuities are fast-- or slowly--moving, and whether the solution is evolved for short or long time. For the special case of smooth solutions, the expected convergence rates are attained for all fluxes and all orders. However, errors are still larger for the simpler fluxes, though differences diminish as the order of accuracy increases. 
    For all selected cases involving discontinuities, differences among fluxes arise for all orders of accuracy considered. Moreover, there are flow situations for which the differences are huge, independently of the order of accuracy of the scheme. The best fluxes are the complete upwind ones. The difference between the best centred flux, FORCE, and incomplete upwind ones is not dramatic, which constitutes and advantage for good centred methods due to their simplicity and generality.