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

Pascalin Tiam Kapen,Tchuen Ghislain

doi:10.1155/2014/827034

Pascalin Tiam Kapen, Tchuen Ghislain

Open Access

https://doi.org/10.1155/2014/827034

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Abstract

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.

Highlights

Today, upwind schemes undoubtedly have become the main spatial discretization techniques used for solving the Euler/Navier-Stokes equations
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)
The performance of the TV-HLL scheme is illustrated in the 1D Euler equations for ideal gas with γ = 1.4

Summary

Introduction

Upwind schemes undoubtedly have become the main spatial discretization techniques used for solving the Euler/Navier-Stokes equations. 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). 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. 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]

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Discussion

Conclusion