FORCE-$\alpha$ Numerical Fluxes within the Arbitrary High Order Semidiscrete WENO-DeC Framework: A Competitive Alternative to Upwind Fluxes

Abstract

This work systematically investigates the performance of FORCE--$\alpha$ numerical fluxes within an arbitrary high order semidiscrete finite volume (FV) framework for hyperbolic partial differential equations (PDEs). Such numerical fluxes have been recently introduced by Toro, Saggiorato, Tokareva, and Hidalgo (Journal of Computational Physics, 416, 2020), and constitute a family of centred fluxes obtained from a suitable modification of First--Order Centred (FORCE) numerical fluxes. In contrast with upwind fluxes, such as Rusanov, Harten--Lax--van Leer (HLL) or the exact Riemann solver (RS) numerical flux, centred ones do not consider in any way the structure of the Riemann problem at cell interfaces. Adopting centred numerical fluxes leads to a high level of flexibility of the resulting numerical schemes, for example in the context of complicated hyperbolic systems, for which RSs may be impossible to construct or computationally expensive. The baseline framework adopted in this investigation is a FV semidiscrete approach with Weighted Essentially Non--Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization, and results are reported up to order 7. Previous investigations involving the same framework have established that increasing the order of accuracy tends to decrease the differences in the results obtained through different numerical fluxes. The goal of this paper is to show that the employment of FORCE--$\alpha$ numerical fluxes within such a framework is a competitive alternative to the adoption of more classical upwind fluxes. The hyperbolic system considered for this investigation is the ideal Euler equations in one and two space dimensions.