On a two-fluid model of two-phase compressible flows and its numerical approximation Mai Duc Thanh Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam article info Article history: Received 22 September 2010 Accepted 7 May 2011 Available online 13 May 2011 Keywords: Two-fluid Two-phase flow Conservation law Source term Lax–Friedrichs Numerical scheme abstract We consider a two-fluid model of two-phase compressible flows. First, we derive several forms of the model and of the equations of state. The governing equations in all the forms contain source terms representing the exchanges of momentum and energy between the two phases. These source terms cause unstability for standard numerical schemes. Using the above forms of equations of state, we construct a stable numerical approximation for this two-fluid model. That only the source terms cause the oscillations suggests us to min- imize the effects of source terms by reducing their amount. By an algebraic operator, we transform the system to a new one which contains only one source term. Then, we discret- ize the source term by making use of stationary solutions. We also present many numerical tests to show that while standard numerical schemes give oscillations, our scheme is stable and numerically convergent. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction We consider in the present paper the stratified flow model for the two-fluid model in one-dimensional space variable. The model with gravity consisting of 6 governing equations is given by (see Staedtke et al. [13] and García-Cascales and Paillère [6]): @ t ða g q g Þþ @ x ða g q g u g Þ¼ 0; @ t ða g q g u g Þþ @ x ða g ðq g u 2 g þ pÞÞ ¼ p@ x a g þ a g q g g; @ t ða g q g E g Þþ @ x ða g ðq g E g þ pÞu g Þ¼p@ t a g þ a g q g u g g; @ t ða l q l Þþ @ x ða l q l u l Þ¼ 0; @ t ða l q l u l Þþ @ x ða l ðq l u 2 l þ pÞÞ ¼ p@ x a l þ a l q l g; @ t ða l q l E l Þþ @ x ða l ðq l E l þ pÞu l Þ¼p@ t a l þ a l q l u l g; ð1:1Þ where a i is the volume fraction, q i is the density, u i is the velocity, e i is the internal energy and E i ¼ e i þ 1 2 u 2 i ; is the total energy, g is the gravity constant in the model with gravity, and g = 0 in the model without gravity, and the sub- script ‘‘i’’ can be ‘‘g’’ or ‘‘l’’, representing the gas or liquid phase of fluids respectively. The volume fractions of the fluid satisfy a g þ a l ¼ 1: 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.05.010 E-mail addresses: mdthanh@hcmiu.edu.vn, hatothanh@yahoo.com Commun Nonlinear Sci Numer Simulat 17 (2012) 195–211 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns