MPS modeling of cross-sectional averaged shallow water flows with open boundaries using TVD-MacCormack predictor-corrector

Abstract

The present study utilizes a moving particle simulation (MPS) method to solve cross-sectional averaged shallow water equations applicable for non-prismatic open channels. The well-known MacCormack predictor–corrector scheme is extended to the Lagrangian framework, ensuring the second accuracy in time and space. For this purpose, upwind and downwind MPS gradient operators are introduced. The total variation diminishing (TVD) flux-limiter scheme is developed to the MPS formulation to preserve monotonicity, which does not require characteristic speeds of the system. In mesh-based methods, the non-conservative form of the momentum equation is problematic in scenarios with discontinuities. However, the non-conservative convective flux term does not exist under the Lagrangian framework in preserving the momentum conservation. Moreover, the cross-sectional area is computed by the algebraic density-ratio equation in the present work since the mass is automatically conserved. This advantage is beneficial for open boundary treatment resulting in less computational effort than mesh-based methods because the partial differential continuity equation is omitted as a boundary value problem. The present high-order solver has positivity-preserving and well-balancing properties without the need for any threshold value for treating the dry bed and friction term. Additionally, it can model free-surface flows in prismatic and non-prismatic open channels with various open and/or closed boundary conditions.