INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2006; 65:834–862 Published online 6 October 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1471 The generalized Riemann problem method for the shallow water equations with bottom topography Jiequan Li 1, ∗, † and Guoxian Chen 1, 2 1 Department of Mathematics, Capital Normal University, Beijing, 100037, People’s Republic of China 2 LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China SUMMARY This paper extends the generalized Riemann problem method (GRP) to the system of shallow water equations with bottom topography. The main contribution is that the generalized Riemann problem method (J. Comput. Phys. 1984; 55(1):1–32) is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes, and at the same step the source term is discretized with an interface method in which only mid-point values are plugged in. This scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method (SGM) (J. Comput. Phys. 2001; 168(1):1–25) into data reconstruction step, and it is also suitable for both steady and unsteady flow simulations. We illustrate the accuracy of this scheme by several 1-D and 2-D numerical experiments. Copyright  2005 John Wiley & Sons, Ltd. KEY WORDS: the shallow water equations; the generalized Riemann problem method; the well- balanced property; the bottom topography; the surface gradient method; characteristic co-ordinates 1. INTRODUCTION The system of shallow water equations (sometimes referred to as the Saint-Venant system) forms the basis of many mathematical models such as the tidal flows in estuary, lakes and coastal water regions, the flood routing in nature and man-made streams, hydraulic jump, ∗ Correspondence to: Jiequan Li, Department of Mathematics, Capital Normal University, Beijing, 100037, People’s Republic of China. † E-mail: jiequan@mail.cnu.edu.cn Contract/grant sponsor: Zheng Ge Ru foundation Contract/grant sponsor: Alexander von Humboldt Foundation Contract/grant sponsor: NNSF; contract/grant number: 10301022 Contract/grant sponsor: Natural Science Foundation Contract/grant sponsor: Fok Ying Tong Education Foundation Contract/grant sponsor: Beijing Educational Commission; contract/grant number: KZ200510028018 Received 20 May 2005 Copyright  2005 John Wiley & Sons, Ltd. Accepted 22 July 2005