Weakly Implicit Numerical Schemes for a Two-Fluid Model

The aim of this paper is to construct semi-implicit numerical schemes for a two-phase (two-fluid) flow model, allowing for violation of the CFL criterion for sonic waves while maintaining a high level of accuracy and stability on volume fraction waves. By using an appropriate hybridization of a robust implicit flux and an upwind explicit flux, we obtain a class of first-order schemes, which we refer to as weakly implicit mixture flux (WIMF) methods. In particular, by using an advection upstream splitting method (AUSMD) type of upwind flux [S. Evje and T. Fl̊atten, J. Comput. Phys., 192 (2003), pp. 175–210], we obtain a scheme denoted as WIMF-AUSMD. We present several numerical simulations, all of them indicating that the CFL-stability of the WIMF-AUSMD scheme is governed by the velocity of the volume fraction waves and not the rapid sonic waves. Comparisons with an explicit Roe scheme indicate that the scheme presented in this paper is highly efficient, robust, and accurate on slow transients. By exploiting the possibility to take much larger time steps, it outperforms the Roe scheme in the resolution of the volume fraction wave for the classical water faucet problem. On the other hand, it is more diffusive on pressure waves. Although conservation of positivity for the masses is not proved, we demonstrate that a fix may be applied, making the scheme able to handle the transition to one-phase flow while maintaining a high level of accuracy on volume fraction fronts.