Keywords: fourth order methods, central schemes, shallow water equations, balanced source term.

The shallow water equations are widely used to model flows in rivers and coastal areas. Numerical methods to solve the shallow water equations show the problem of false numerical oscillations and excessive numerical diffusion that can occur around the discontinuities of the solution. Thus, it is necessary to develop efficient numerical methods to solve this problem accurately. Classical numerical methods, of the type upwind, require the calculation of flows in the boundary of the cells in which the solution is integrated. Central schemes avoid this problem so the discontinuities in the pointwise solution, produced by the reconstruction algorithm, are located at the center of the staggered control volumes, allowing a simpler reconstruction of the numerical fluxes [1].

Balaguer and Conde [2] describe a new fourth-order non-oscillatory central scheme for solving hyperbolic conservation laws. Time integration is performed using a Runge-Kutta scheme with a natural continuous extension. For the spatial reconstruction it calculates the point values of the solution from the cell averages by avoiding the increase in the number of solution extrema at the interior of each cell. This operator also guarantees that the number of extrema does not exceed the initial number of extrema. This property makes the numerical solution show a non-oscillatory behaviour. This scheme has been applied to solve accurately various problems with non-linear fluxes and the Euler equations of gas dynamics.

In this paper, we are concerned with the construction of high order well balanced non-oscillatory finite volume schemes for solving the shallow water equations with a non-flat bottom topology. We present an extension of the central non-oscillatory scheme given in [2] to solve the shallow water equations over a movable non-flat bed described in [3]. Time integration is obtained following a Runge-Kutta procedure, coupled with its natural continuous extension (NCE). Spatial accuracy is obtained with three-degree reconstruction polynomials in each cell, keeping the local monotonicity of the interpolation data. We use the treatment for the bed slope source term described in Caleffi et al. [1], which maintains the established order of accuracy and satisfies the exact conservation property (C-property).

Several standard one-dimensional test cases are used to verify behaviour of our scheme and its non-oscillatory properties.

1

V. Caleffi, A. Valiani, A. Bernini, "High-order balanced CWENO scheme for movable bed shallow water equations", Advances in water resources, 30, 730-741, 2007. doi:10.1016/j.advwatres.2006.06.003

2

A. Balaguer, C. Conde, "Fourth-Order Nonoscilatory Upwind and Central Schemes for Hyperbolic Conservation Laws", SIAM Journal on Numerical Analysis, 43(2), 455-473, 2006. doi:10.1137/S0036142903437106

3

N. Crnjaric-Zic, S. Vukovic, L. Sopta, "Extension of ENO and WENO schemes to one-dimensional sediment transport equations", Computers & Fluids, 33, 31-56, 2004. doi:10.1016/S0045-7930(03)00032-X

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