Fourth order balanced WENO schemes for shallow water equations

Two WENO schemes, fourth-order accurate in space and time, for the numerical integration of shallow water equations with bottom slope source term, are presented. Spatial accuracy is achieved using WENO reconstructions of the free-surface elevation and of the specific discharge. In the first Central WENO model, based on staggered grids, time accuracy is achieved using a Runge-Kutta (RK) scheme coupled with its Natural Continuous Extension (NCE). In the second Upwind WENO model, time accuracy is obtained using a Strong Stability Preserving Runge-Kutta method, SSPRK(5,4). Original source term treatment, satisfying the C-property, i.e. the property of exactly preserving the quiescent flow, is used in both the models. Such a treatment involves the spatial integration of the source term, which is analytically manipulated in order to take advantage from the regularity of the free surface elevation, usually smoother than the bottom elevation. The source term treatment in the Central scheme involves also an original evaluation of the point-values of the flux gradient, coupled with the source term. Several standard one-dimensional test cases are used to verify the high-order accuracy, the C-property, and the good resolution properties of both the models. Both the schemes are reliable for the application to real problems characterized by very irregular bottoms