ENO and WENO Schemes with the Exact Conservation Property for One-Dimensional Shallow Water Equations

In this paper we propose new finite difference numerical schemes for hyperbolic conservation law systems with geometrical source terms. In the development of the new schemes we use the essentially nonoscillatory (ENO) and weighted ENO (WENO) reconstruction, developed by Harten, Osher, Engquist, Chakravarthy, Shu, and Jiang, and the idea of the balancing between the flux gradient and the source term, introduced by Bermùdez and Vلzquez. Actually, the new schemes are ENO and WENO schemes with the source term decomposed, i.e., the ENO and WENO reconstruction is applied not only to the flux but to a combination of the flux and the source term. In particular, when new schemes are applied to the shallow water equations the new schemes verify the exact conservation property (C-property). We present the algorithm, the proof of the exact C-property, and numerical results for several test problems.