A New Flux Splitting Scheme Based on Toro-Vazquez and HLL Schemes for the Euler Equations

This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012) and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro. 1. Introduction Today, upwind schemes undoubtedly have become the main spatial discretization techniques used for solving the Euler/Navier-Stokes equations. Indeed, an interesting feature of these schemes is that the discretization of the equations on a mesh is performed according to the direction of propagation of information on that mesh. They are categorized [1] into two major family schemes, namely, flux vector splitting (FVS) and flux difference splitting (FDS). Concerning the flux difference splitting schemes (FDS), they are based on the difference in the decomposition of fluxes, constructed on an approximated solution of the local Riemann problem between two adjacent states. There are several FDS schemes formulated in the literature among which are the Roe and HLL numerical schemes. The HLL scheme developed by Harten-Lax-Van Leer is a direct approximation of the numerical flux to compute Godunov flux [2]. It has been shown that the HLL scheme is very efficient and robust, has an entropy satisfaction property, resolves isolated shock exactly, and preserves positivity [2–4]. Unfortunately, the main demerit of the HLL scheme is that it cannot resolve contact discontinuity exactly. Another well-known FDS scheme is Roe’s scheme [5] largely used because of its accuracy, quality, and mathematical clarity. Unfortunately, Roe’s scheme admits rarefaction shocks that do not satisfy the entropy condition and sometimes gives rise to spurious solutions such as carbuncle phenomena and odd-even decoupling [6]. To improve the quality of solutions, Quirk proposed a strategy to use combined fluxes so that a dissipative approach can be used in the shock regions [7]. In contrast, the flux vector splitting (FVS) has proven to be a simple and useful technique for arriving at upwind differencing and is