Hybrid fifth-order unequal-sized weighted essentially non-oscillatory scheme for shallow water equations

In this paper, we propose a new discontinuous sensor and a finite difference hybrid unequal-sized weighted essentially non-oscillatory (WENO) scheme with fifth-order accuracy for solving shallow water equations with or without source terms. The developed discontinuous sensor is directly designed based on the highest degree polynomial obtained from the five-point stencil in the unequal-sized WENO procedures, and can automatically identify the discontinuous region without manually adjusting the parameters related to the problem. Subsequently, a hybrid unequal-sized WENO scheme is developed through this newly designed discontinuous sensor, which uses the cheap linear scheme in the smooth regions and the existing expensive unequal-sized WENO scheme in the vicinity of discontinuous, thus achieving the goal of inheriting the excellent characteristics of the unequal-sized WENO scheme while reducing its computing time. Finally, some benchmark numerical examples are provided to verify the performance of this new WENO scheme for solving shallow water equations in terms of high-order accuracy, exact conservation property, shock capture capability, and computational efficiency.