An unstructured second-order subgrid method for the shallow water equations

Abstract: We propose a new unstructured numerical subgrid method for solving the shallow water equations using a finite volume method with enhanced bathymetry resolution. The method employs an unstructured triangular mesh with support for triangulation of elements to form finer subgrids locally. The bathymetry is represented on the fine mesh, allowing the incorporation of small-scale features, while velocities are defined on the coarse mesh. The governing equations are solved numerically on the coarse mesh, making the method computationally cheaper compared to a traditional fine mesh computation. To accurately represent the velocities, we employ a second-order accurate WENO method and for temporal integration an explicit second-order accurate Runge-Kutta method. Furthermore, we present a novel subgrid face value reconstruction that accounts for partially wet cells, where only some of the subgrid cells are wet. Together with a newly developed gravity source term discretization, we demonstrate that the scheme is well-balanced on the subgrid level. Finally, the subgrid method is validated on several test cases to show: i) that the scheme is well-balanced, ii) confirm the order of the accuracy of the scheme, and iii) demonstrate the capability of handling moving flood and dry boundaries. We also highlight when it is beneficial to increase the number of subgrid cells to improve the results of the finite volume method.