Hybrid flux-splitting schemes for a common two-fluid model

The aim of this paper is to construct hybrid flux vector splitting (FVS) and flux difference splitting (FDS) schemes for a commonly used two-fluid model consisting of two separate momentum equations. This is done by refining ideas previously applied to develop hybrid FVS/FDS schemes for a simpler two-phase model consisting of a mixture momentum equation [J. Comput. Phys. 175 (2002) 674]. More specifically, we seek to construct upwind type of schemes which are not based on calculations of the full eigenstructure of Jacobi matrices as needed by approximate Riemann solvers like the Roe scheme. Based on a crude approximation of the eigenstructure of the model, we derive schemes of the van Leer and FVS type. We demonstrate that these schemes possess desirable stability properties, but are excessively diffusive. By adapting ideas originally suggested by Wada and Liou [SIAM J. Sci. Comput. 18 (1997) 633] for the Euler equations, we suggest a mechanism for removing numerical dissipation. We present numerical simulations where we compare the performance of the resulting schemes with that of the Roe scheme, and by that shed light on the issues of accuracy, efficiency, and robustness of the proposed schemes. Particularly, we consider the classical water faucet problem as well as a stiff separation problem which locally involves transition from two-phase to single-phase flow. Results from these test cases show that we are able to construct hybrid FVS/FDS schemes which properly combine the accuracy of FDS in the resolution of sharp mass fronts and the robustness of FVS which ensures stability under stiff conditions.