Using natural element mesh-free numerical method in solving shallow water equations

Abstract

This article attempts to solve shallow water equations based on the natural element method (NEM). The foundation of the NEM method is based on the natural neighbour interpolation, and, in this article, the Sibson interpolation is used. Moreover, the nodal integration is applied for discretising the governing equations. The proposed method is validated by three 2D shallow water test cases, including (1) supercritical flow in converged channel, (2) subcritical flow in sudden expansion and (3) inflow jet into the main channel, and then compared with the experimental data. In addition, in order to evaluate the obtained results, the outcomes of the models are compared with the results of the conventional finite volume method (FVM). The results are evaluated by NRMSE and relative error criteria. In general, results indicate that the proposed NEM method is slightly superior to the FVM.

Keywords:

• shallow water equations
• natural element method
• natural neighbour interpolation
• nodal integration
• supercritical flow
• Lax–Friedrich scheme