A well-balanced and positivity-preserving SPH method for shallow water flows in open channels

A well-balanced and positivity-preserving meshless method based on smoothed particle hydrodynamics (SPH) is developed to simulate one-dimensional (1D) and two-dimensional (2D) shallow water (SW) flows in open channels with irregular geometries. A new form of the characteristic equations that govern the water-surface level and water velocity is introduced to specify the numerical inflow/outflow boundary conditions. An additional condition, derived from temporal discretization to determine the time-step size, forces the water depth to be positive. A 1D finite volume shallow water (FVSW) model based on the first-order Godunov upwind method is built to conduct a comparison of the 1D meshless-based and mesh-based SW models. Six benchmark cases - still water, single trapezoidal and rectangular and prismatic and non-prismatic channels, and a dendritic channel network - are employed to validate the proposed models and compared with the exact and mesh-based numerical solutions. A real-world case of the Chicago Area Waterways System (CAWS) is investigated to highlight the performance of the proposed 1D model for a practical hydraulic system.