A shock-capturing meshless method for solving the one-dimensional Saint-Venant equations on a highly variable topography

The Saint-Venant equations are numerically solved to simulate free surface flows in one dimension. A Riemann solver is needed to compute the numerical flux for capturing shocks and flow discontinuities occurring in flow situations such as hydraulic jump, dam-break wave propagation, or bore wave propagation. A Riemann solver that captures shocks and flow discontinuities is not yet reported to be implemented within the framework of a meshless method for solving the Saint-Venant equations. Therefore, a wide range of free surface flow problems cannot be simulated by the available meshless methods. In this study, a shock-capturing meshless method is proposed for simulating one-dimensional (1D) flows on a highly variable topography. The Harten–Lax–van Leer Riemann solver is used for computing the convective flux in the proposed meshless method. Spatial derivatives in the Saint-Venant equations and the reconstruction of conservative variables for flux terms are computed using a weighted least square approximation. The proposed method is tested for various numerically challenging problems and laboratory experiments on different flow regimes. The proposed highly accurate shock-capturing meshless method has the potential to be extended to solve the two-dimensional (2D) shallow water equations without any mesh requirements.