SIAM J. SCI. COMPUT. c  2009 Society for Industrial and Applied Mathematics Vol. 31, No. 3, pp. 1742–1773 CENTRAL-UPWIND SCHEMES FOR TWO-LAYER SHALLOW WATER EQUATIONS ∗ ALEXANDER KURGANOV † AND GUERGANA PETROVA ‡ Abstract. We derive a second-order semidiscrete central-upwind scheme for one- and two- dimensional systems of two-layer shallow water equations. We prove that the presented scheme is well-balanced in the sense that stationary steady-state solutions are exactly preserved by the scheme and positivity preserving; that is, the depth of each fluid layer is guaranteed to be nonnegative. We also propose a new technique for the treatment of the nonconservative products describing the momentum exchange between the layers. The performance of the proposed method is illustrated on a number of numerical examples, in which we successfully capture (quasi) steady-state solutions and propagating interfaces. Key words. hyperbolic systems of conservation and balance laws, semidiscrete central-upwind schemes, nonconservative products, two-layer shallow water equations AMS subject classifications. 76M12, 35L65, 76T99 DOI. 10.1137/080719091 1. Introduction. We develop a Godunov-type central-upwind scheme for the system of two-layer shallow water equations. This system is obtained from the com- pressible isentropic Euler equations by vertical averaging across each layer depth. The layers are assumed to have different constant densities ρ 1 <ρ 2 due to, for example, different water salinity, and to be immiscible. The studied one- and two-dimensional two-layer shallow water systems are extensions of the Saint-Venant systems [11], which are widely used in both geophysical science and coast and dams-keeping engineering. The one-dimensional (1-D) two-layer shallow water model describes a flow that consists of two layers of heights h 1 (upper layer) and h 2 (lower layer) at position x at time t with corresponding velocities u i and discharges q i := h i u i ,i =1, 2. The two-layer system we consider is the model studied in [5], which describes a flow in a straight channel with a bottom topography B. It is given by ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (h 1 ) t +(q 1 ) x =0, (q 1 ) t + ( h 1 u 2 1 + g 2 h 2 1 ) x = -gh 1 B x - gh 1 (h 2 ) x , (h 2 ) t +(q 2 ) x =0, (q 2 ) t + ( h 2 u 2 2 + g 2 h 2 2 ) x = -gh 2 B x - gh 2   h 1  x , (1.1) where g is the gravitational constant, r := ρ1 ρ2 is the constant density ratio, and  h 1 := rh 1 . The 2-D generalization of (1.1) is (for details see, e.g., [28]) the following system ∗ Received by the editors March 22, 2008; accepted for publication (in revised form) November 10, 2008; published electronically February 27, 2009. http://www.siam.org/journals/sisc/31-3/71909.html † Mathematics Department, Tulane University, New Orleans, LA 70118 (kurganov@math.tulane. edu). The work of this author was supported in part by NSF grant DMS-0610430. ‡ Department of Mathematics, Texas A & M University, College Station, TX 77843 (gpetrova@ math.tamu.edu). The work of this author was supported in part by NSF grants DMS-0505501 and DMS-0810869 and by award KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST). 1742