An Integrated Media, Integrated Processes Watershed Model – WASH123D: Part 4 – A characteristics-based finite element method for 2-D overland flow

Abstract

The Method of Characteristics (MOC) in the context of finite element method was applied to the complete 2-D shallow water equations for 2-D overland flow. For two- dimensional overland flow, finite element or finite volume methods are more flexible in dealing with complex boundary. Recently, finite volume methods have been very popular in numerical solution of the shallow water equations. Some have pointed out that finite volume methods for 2-D flow are fundamentally one-dimensional (normal to the cell interface). The results may rely on the grid orientation. The search for genuinely multidimensional numerical schemes for 2-D flow is an active topic. We consider the Method of Characteristics (MOC) in the context of finite element method as a good alternative. Many researchers have pointed out the advantage of MOC in solving 2-D shallow water equations that are of the hyperbolic type that has wave- like solutions and at same time, considered MOC for 2-D overland flow being non- tractable on complex topography. The intrinsic difficulty in implementing MOC for 2- D overland flow is that there are infinite numbers of wave characteristics in the 2- D context, although there only three independent wave directions are needed for a well-posed solution to the characteristic equations. We have implemented a numerical scheme that attempts to diagonalize the characteristic equations based on pressure and velocity gradient relationship. This new scheme was evaluated by comparison with other choice of wave characteristic directions in the literature. Example problems of mixed sub-critical flow/super-critical flow in a channel with approximate analytical solution was used to verify the numerical algorithm. Then experiments of overland flow on a cascade of three planes (Iwagaki 1955) were solved by the new method. The circular dam break problem was solved with different selections of wave characteristic directions and the performance of each selection was evaluated based on accuracy and numerical stability. Finally, 2-D overland flow over complex topography in a wetland setting with very mild slope was solved by the new numerical method to demonstrate its applicability.