1-D WENO scheme with the exact conservation property for open-channel flow simulation

Shallow water equations have been usually used to describe the flow fields in rivers, lakes, estuaries, and seashores. Shallow water equations are one of the hyperbolic-type PDE's in mathematical realm. Traditional finite difference methods cannot have high accuracy of the solution in smooth regions and without numerical oscillations at the singularities at the same time. They usually cause the simulation results having numerical damping, numerical oscillation, and not satisfying the conservation law. Solving the hyperbolic-type equations by WENO has two significant advantages: the solution can describe the phenomena clearly in the region with singularities. Besides, WENO scheme is easy to apply to two-dimensional system, three-dimensional system and even to flow fields with complicated source terms. Considering all these advantages, this paper combines conservative WENO with finite volume method for solving the one dimensional shallow water equations. With the modification of determing characteristic velocity, the original scheme can be improved to deal with the dry-bed problem and the rarefaction wave problem. Moreover, by deducing mathematical differential equations, the combination of WENO and finite volume method can be proved to satistify the conservation law. Finally, this paper demonstrates the validity of the proposed scheme by simulating several laboratory experiments, e.g., flow over a bump, dam break, rarefaction wave, and dry bed problems.