An accurate, robust and efficient convection-pressure flux splitting scheme for compressible Euler flows

Abstract

The popular Roe's flux difference splitting scheme has high resolution for various types of discontinuities but it fails to satisfy the entropy condition and will encounter different forms of instability while calculating shock waves, such as the post-shock oscillation of the slowly moving shock waves and the carbuncle phenomenon of the multidimensional strong shock waves. In the present work, a simple flux difference splitting scheme based on the idea of convection-pressure splitting is proposed. With application of the Toro-Vázquez splitting procedure, the flux of the 3D Euler equations is split into the convection part and the pressure part. Due to having a complete set of linearly independent eigenvectors, the pressure subsystem can be solved by the conventional flux difference splitting method. The weakly hyperbolic convection subsystem, however, does not have a complete set of linearly independent eigenvectors and thus is solved by the flux difference splitting scheme based on the generalized eigenvectors [1]. In addition, a strategy that combines the THINC reconstruction with the BVD algorithm is employed on the numerical dissipation term of the convection flux to further improve the resolution for contact discontinuities. A comprehensive range of numerical simulations for classical 1D, 2D and 3D benchmark test problems demonstrate the good performance of the proposed scheme.