Solving the Zero-Inertia/Diffusive-Wave Hydrodynamic Model as a Nonlinear Advection Problem with the Finite Element Method: Preliminary Results

The zero-inertia/diffusive-wave hydrodynamic model is often numerically solved as a nonlinear parabolic/diffusion differential problem, similar to the heat equation. In this approach, the function of the sought solution usually represents the elevation of the water surface of the flow. Such numerical solutions have been implemented using the finite difference, finite volume, or finite element method and are applied in the estimation of river flow, watershed runoff, or flood inundation, for example.

In this study, the two-dimensional zero-inertia/diffusive-wave hydrodynamic model was solved as a nonlinear advection differential problem. The function of the sought solution is the thickness (sometimes called flow height) of the water flow. An implicit finite difference scheme was applied for the time domain. In order to handle irregular geometries, the Galerkin finite element method was employed. To enhance the stability of the numerical solution, stabilization terms were added to the numerical scheme, namely, the streamline-upwind Petrov-Galerkin (SUPG) term and the spurious-oscillations-at-layers diminishing (SOLD) term. Different types of boundary conditions at the outflow boundaries were imposed.

When simple tests were carried out, the results were in good agreement with the analytical or some existing numerical solutions. While more complicated tests were conducted, the results were still realistic, although with defects.

The presented approach can be used for numerical solutions with longer time domains. Further investigations are needed to improve the numerical solution in the frame of this approach. Future studies can help in choosing better numerical schemes for the time domain and other stabilization terms in the finite element method.