We present a twedimensional, well-balanced, centrai-upwind scheme for approximating SdVtiOPS of the s h d !~%~ Water equations in thz presence Uf d statiuaary buttvrri lvpography on triangular meshes. Our starting point is the recent central scheme of Kurganov and Petrova (KP) for approximating solutions of conservation laws on triangular meshes. In order to extend this scheme from systems of conservation laws to systems of balance laws one has to find an appropriate discretization of the

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... terms. We first show that for general triangulations there is no discretization of the source terms that corresponds to a well-balanced form of the KP scheme. We then derive a new variant of a central scheme that can be balanced on We note in passing that it is straightforward to extend the K P scheme to general unstructured conformal meshes. This extension allows us to recover our previous well-balanced scheme on Cartesian grids. We conclude with several simulations, verifying the second-order accuracy of our scheme as well a s its well-balanced properties. tri2Erni 137 mneh n c U U U L A b . , . i 1 cunsider a flonin a two-dimensional channel u-ith a bottom elevation given by B ( 2 ) , where 2 = (x. y) Let H ( Z , t ) represent the fluid depth abot-e the bottom, and C(Z. t ) = (u(Z, t ) . ~( 2 . t ) ) be the fluid velocity. The top surface at any time t is denoted by w(Z, t ) = B ( Z ) + H ( 2 . t ) The shallow water equations, introduced by Saint-Venant, in [22] , are commonly used t.o model f l o~s in rivers or coastal areas. U'hen mTritt,en in terms of the t.op surface u and the momentum 'Program in Scientific CornputingiComputational Mathemakics, Stanford UniversitJ-and t.he X.2S.4 Advanced Supexonpuiing Division: K-4S-k Ames &search Center, MoEett Field; CA 94035-1000: br~-son:Bnas.nasa.gov TDepart,ment, of M;i.t.hem.ptics: St.~∨? tiI'IVPrsity> St~nfnrct. C-4 4430-21 25; tl!esn.~:~;:m?"~h.stI??f"rd.ec',n f (Hu, "L:) these equations are of the form wt + (Hu), + (He), = 0, (Hu) (Hv) Y This choice of variables is particularly suitable for dealing with stationary steady-state solutions (see [la, 211 for details). For simplicity we fix the gravitational constant, 9, from now on to be g = 1. In this work we are interested in approximating solutions of (1.1) on triangular meshes. Our goal is to investigate how to adapt the semi-discrete central schemes on triangular meshes that were recently introduced by Kurganov and Petrova in [14] to this problem. We are interested in derivihg a discretization of the source terms in (1.1) that preserves stationary steady-state solutions, as such solutions play an important role in the dynamics of (1.1). Central schemes for conservation laws have become popular in recent years a s a tool for approximating solutions for multi-dimensional systems of hyperbolic conservation laws. Like other Godunov-type schemes, central schemes are based on a three-step procedure: a reconstruction step in which an interpolant is reconstructed from previously computed cell-averages; an evolution step in which this interpolant is evolved exactly in time according to the equations; and a projection step in which the solution is projected back to cell-averages. When compared with other methods, central schemes are particularly appealing since they do not require any Riemann solvers and systems can be solved component-wise. A first-order prototype of central schemes is the Lax-Friedrichs scheme [6]. A second-order extension is due to Nessyahu and Tadmor [19]. Extensions to two dimensions are due to Arminjon, Jiang and Tadmor [l, 111. By estimating bounds on the local speeds of propagation of information from discontinuities, it is possible t o pass to the semi-discrete limit (see [13, 151 and the references therein). There are several extensions of central schemes to unstructured grids. A fully discrete method is due to Arminjon et al. [2]; a recent semi-discrete scheme was proposed by Kurganov and Petrova in [14I. Balanced Central schemes for shallow water equations on Cartesian grids are due to Russo in the fully-discrete framework [2l] (see also [18]) and to Kurganov and Levy in the semi-discrete framework [12]. There are many approaches to approximating solutions of (1.1). VJe refer, e.g. to [3, 4, 5, 7, 9, 16, 17, 201 and the references therein. Our goal in this paper is to show that balancing is also possible with central schemes. We would like to emphasize that this is tie-firsf time. in-cfhich the balancing issues are treated for central schemes on unstructured grids. The paper is organized as follows. We start in Section 2 with a brief overview of the K P central scheme on triangular meshes. We note that this scheme is not limited to triangular meshes and it can be equally well applied to general unstructured grids. We also make the necessarj-adjustments to incorporate source terms into the scheme. We proceed in Section 3 -with the cliscreTization of the cell-averages of the Source terms for t.he shallow-water equations. Thz g o d is tc Ex! s discretizitinn slxh tjhzi. t41e s~h e m~ will preserve scationar?; st,ead!-stat,es.