Improvements of semi-implicit schemes for hyperbolic balance laws applied on open channel flow equations

Abstract

In this paper, we consider a well-balanced semi-implicit one parameter family of schemes. The presented schemes are a generalization of the well-balanced upwind explicit finite volume schemes. The schemes are applied to the Saint Venant open channel flow equations. The main feature of the presented schemes is their balancing property, achieved even for the channels with the general cross section geometry. In the paper, we present the scheme algorithm and the proof of the exact conservation property when the proposed schemes are applied to the open channel flow equations. Furthermore, the schemes’ accuracy and stability are improved by using a local semi-implicit approach, which takes into account the CFL number locally. In this way, the highly efficient, robust, and unconditionally stable family of balanced numerical schemes is developed. Newly developed schemes are able to give accurate low diffusion results in stationary as well as in non-stationary test cases. Since particular attention is focused on the simulation efficiency on real engineering problems, an algorithm for treatment and precomputation of channel geometry parameters is presented. The algorithm significantly reduces the computational cost of the simulation. Behavior of new schemes is analyzed in several idealized test cases, and on a simulation of a realistic flood wave propagation in the Kupa river involving friction, non-uniform bed slopes and strong channel width variations.