Artificial Viscosity Stabilization of Meshless Methods for Shallow Water Flows in Channels

The purpose of this paper is to present an efficient localized meshless methods based on radial basis functions to accurately analyze the shallow water equations (SWEs) in open channels. It is a hyperbolic system of first-order nonlinear partial differential equations. Due to their classification as advection equations, they allow for discontinuous solutions, often characterized by shock behaviour in simulations. Therefore, the development of an efficient and accurate numerical model for analyzing the SWEs holds critical importance in scientific research. To address non-physical oscillations near discontinuities, we developed a technique involving artificial viscosity, which is integrated with the local meshless methods for spatial discretization of the SWEs, while temporal discretization is achieved using the fourth-order Runge-Kutta method. A series of experiments has been conducted to evaluate the effectiveness and accuracy of the suggested methods. These experiments included examining flow at a steady state with and without friction, as well as studying the dam-break scenario on a wet bed. The results compared with analytical solutions, demonstrate that the artificial viscosity combined with the proposed methods proficiently capture shocks and effectively handles discontinuous flow by introducing an appropriate viscosity coefficient into the equations. Overall, the results are satisfactory and show good agreement.