Central unstaggered finite volume methods for shallow water equations

Abstract

In this paper we develop a new central unstaggered finite volume method for solving systems of hyperbolic equations. Based on the Lax‐Friedrichs central scheme and on the Nessyahu and Tadmor (NT) one‐dimensional non‐oscillatory central scheme, we construct a new class of unstaggered second‐order non‐oscillatory central finite volume schemes for approximating solutions of hyperbolic systems of conservation laws. In contrast with the original (NT) central scheme that evolves the numerical solution on an original grid (at even time steps) and on a staggered one (at odd time steps), the method we propose evolves the numerical solution on a single grid and uses a “ghost” staggered grid to avoid the time consuming resolution of the Riemann problems arising at the cell interfaces. The numerical solution is defined on the computational domain using piecewise linear interpolants. To avoid undesired oscillations a slope limiting process is applied; this results in a scheme that is second‐order accurate in space. To guarantee second‐order accuracy in time a second‐order quadrature rule is applied. We apply our numerical scheme and solve some classical shallow water equation problems. The numerical results presented in this work show the efficiency and the potential of our unstaggered central scheme; they compare very well with those obtained using the original (NT) central scheme and are in a very good agreement with corresponding results appearing in the recent literature.