Implicit 2D surface flow models performance assessment: Shallow Water Equations vs. Zero-Inertia Model

Abstract

Zero-Inertia (ZI) models are used in overland flow simulation due to their mathematical simplicity, compared to more complex formulations such as Shallow Water (SW) models. The main hypothesis in ZI models is that the flow is driven by water surface and friction gradients, neglecting local accelerations. On the other hand, SW models are a complete dynamical formulation that provide more information at the cost of a higher level of complexity. In realistic problems, the usually huge number of cells required to ensure accurate spatial representation implies a large amount of computing effort and time. This is particularly true in 2D models. Hence, there is an interest in developing efficient numerical methods. In general terms, numerical schemes used to solve time dependent problems can be classified in two groups, attending to the time evaluation of the unknowns: explicit and implicit methods. Explicit schemes offer the possibility to update the solution at every cell from the known values but are restricted by numerical stability reasons. This can lead to very slow simulations in case of using fine meshes. Implicit schemes avoid this restriction at the cost of generating a system of as many equations as computational cells multiplied by the number of variables to solve. In this work, an implicit finite volume numerical scheme has been used to solve the 2D equations in both ZI and SW models. The scheme is formulated so that both quadrilateral and triangular meshes can be used. A conservative linearization is done for the flux terms, leading to a non-structured matrix for unstructured meshes thus requiring iterative methods for solving the system. A comparison between 2D SW and 2D ZI is done in terms of performance, efficiency and mesh requirements, in which both models benefit of an implicit temporal discretization in steady and nearly-steady situations.