A meshless geometric conservation weighted least square method for solving the shallow water equations

TL;DR

This paper introduces a novel meshless numerical method that combines geometric conservation and weighted least squares to effectively solve shallow water equations, including shock capturing, on irregular geometries.

Contribution

The paper presents the first shock-capturing meshless method for shallow water equations using a geometric conservation weighted least squares approach with integrated Riemann solver.

Findings

Successfully captures shocks in open-channel flows

Demonstrates high accuracy on analytical and experimental tests

Effective on large-scale dam-break simulations

Abstract

The shallow water equations are numerically solved to simulate free surface flows. The convective flux terms in the shallow water equations need to be discretized using a Riemann solver to capture shocks and discontinuity for certain flow situations such as hydraulic jump, dam-break wave propagation or bore wave propagation, levee-breaching flows, etc. The approximate Riemann solver can capture shocks and is popular for studying open-channel flow dynamics with traditional mesh-based numerical methods. Though meshless methods can work on highly irregular geometry without involving the complex mesh generation procedure, the shock-capturing capability has not been implemented, especially for solving open-channel flows. Therefore, we have proposed a numerical method, namely, a shock-capturing meshless geometric conservation weighted least square (GC-WLS) method for solving the shallow water equations. The HLL (Harten-Lax-Van Leer) Riemann solver is implemented within the framework of the proposed meshless method. The spatial derivatives in the shallow water equations and the reconstruction of conservative variables for high-order accuracy are computed using the GC-WLS method. The proposed meshless method is tested for various numerically challenging open-channel flow problems, including analytical, laboratory experiments, and a large-scale physical model study on dam-break event.