SIAM J. SCI. COMPUT. c  2009 Society for Industrial and Applied Mathematics Vol. 31, No. 4, pp. 2866–2889 SOLUTION OF THE SEDIMENT TRANSPORT EQUATIONS USING A FINITE VOLUME METHOD BASED ON SIGN MATRIX ∗ FAYSSAL BENKHALDOUN † , SLAH SAHMIM ‡ , AND MOHAMMED SEAID § Abstract. We present a finite volume method for the numerical solution of the sediment trans- port equations in one and two space dimensions. The numerical fluxes are reconstructed using a modified Roe scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the sediment transport system. A well-balanced discretization is used for the treatment of source terms. The method is well balanced, nonoscillatory, and suitable for both structured and unstructured trian- gular meshes. An adaptive procedure is also considered for the two-dimensional problems to update the bed-load accounting for the interaction between the bed-load and the water flow. The proposed method is applied to several sediment transport problems in one and two space dimensions. The numerical results demonstrate high resolution of the proposed finite volume method and confirm its capability to provide accurate simulations for sediment transport problems under flow regimes with strong shocks. Key words. shallow water equations, sediment transport problems, finite volume method, well-balanced discretization, unstructured mesh AMS subject classifications. 35L50, 76M12, 74J15 DOI. 10.1137/080727634 1. Introduction. In recent years significant advances have been made in com- putational shallow water flows on a fixed bed. The transport of sediment by shallow water flows is an important phenomenon that occurs in many river and coastal engi- neering applications. These applications include propagation of floods inside gullies, change of the beach profile due to severe wave climates, torrents or rivers after heavy rain, response of the seabed to dredging schemes or imposed structures, or movement induced by the abrupt collapse of barriers. Typically, research on river processes was based on field observations and laboratory scale modeling. The measurements from the fields are usually costly and difficult to realize especially for large-scale sediment transport systems. On the other hand, the laboratory research has been essentially used as design and verification tools for understanding complex river processes, despite its high cost of construction, maintenance, and operation. Nowadays, an alternative is numerical modeling of sediment transport, which has attracted many research stud- ies. It should be stressed that numerical simulation of morphodynamic changes of the bed in hydraulic systems involves different physical mechanisms propagating within the system according to their time response; i.e., the problem is a multiscale system which requires a robust solver to accurately resolve both hydrodynamic and morpho- dynamic time scales. In addition, most morphodynamical flows frequently involve important features such as internal bores and moving shoreline fronts which present a significant challenge to the accuracy and stability of numerical models. The objec- ∗ Received by the editors June 17, 2008; accepted for publication (in revised form) April 6, 2009; published electronically July 3, 2009. This work was supported by CMLA, ENS Cachan. http://www.siam.org/journals/sisc/31-4/72763.html † LAGA, Universit´ e Paris 13, 99 Av J.B. Clement, 93430 Villetaneuse, France (fayssal@math. univ-paris13.fr), and CMLA, ENS Cachan, 61 avenue du Pdt Wison, 94 235 Cachan, France. ‡ Institut sup´ erieur d’informatique et du Multim´ edia de Sfax, Route M’Harza 1.5 Km, BP 1030, 3018 Sfax, Tunisia (sahmim slah@yahoo.fr). § School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK (m.seaid@ durham.ac.uk). 2866