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Balanced finite volume WENO and central WENO schemes for the shallow-water and the open-channel flow equations

This paper deals with the extensions of the finite volume WENO schemes and of the central WENO schemes to the shallow-water and to the open-channel flow equations. In both considered balance laws the source term depending on the properties of the channel arises. The decomposed approach used in its numerical evaluation, which maintains the balancing between the flux gradient and the source term, becomes crucial for obtaining stable numerical schemes that are free of spurious oscillations. In addition, the open-channel flow equations contain spatially variable flux function. The appropriate definitions of the terms that arise in the decomposition of the source term in combination with the Roe approximate solver that includes the spatial derivative of the flux function, lead to the finite volume WENO scheme that satisfies the exact conservation property - the property of preserving the quiescent flow exactly. When the central WENO schemes are applied additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vise a versa by using the WENO reconstruction procedure. The proposed central WENO schemes also preserve the quiescent flow, but only in prismatic channels. In various test problems the obtained balanced schemes show significant improvement comparing with the standard versions of the schemes.

finite volume WENO schemes; central WENO schemes; shallow-water equations; open-channel flow equations; balanced scheme; exact conservation property; hyperbolic balance law

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