A new method for moving–boundary hydrodynamic problems in shallow water

Dynamic moving–boundary problems are a common occurrence in shallow–water hydraulics, yet no generally applicable or computationally efficient framework is available for their solution. In this paper a new scheme is presented that attempts to solve the problem of representing moving–boundary shallow–water hydrodynamics on fixed numerical grids. The scheme consists of three parts: identification of partly wet elements; the development of physically appropriate treatments to deal with mass and momentum conservation discrepancies in such areas within a two–dimensional finite–element framework; and the development and testing of a proof of mass conservation for the new scheme. In particular, the algorithm is unique in distinguishing between flooding and dam–break partly wet elements in contrast to previous solutions to this problem which typically treat both types in a similar fashion. In reality, consideration of the problem physics shows that mass and momentum conservation discrepancies can only be positively identified and corrected on elements of the flooding type. Accordingly, this disaggregation of partly wet elements into flooding and dam–break types is the approach adopted. This new scheme is tested against a structured series of numerical experiments including objective tests, a newly developed analytical solution for this problem and a unique high–resolution topographic data–set recently collected for an area of tidal beach on the eastern coast of the UK. The new scheme is shown to be physically realistic, mass conservative and appears to offer a significant improvement over standard finite–element techniques.