Improving the wetting-drying algorithm of a finite-element, shallow-water model

The earth is covered of 71% of water including fluvial and estuarine regions that represent less than 1% of the hydrosphere. However, those regions are the main actors for their respective ecosystems and have a significant amount of biomass compared to the oceans. Thus, it is natural that we build software that allow us to make predictions about the behaviour of the fluid in the different locations of our planet. For example, we can quote the SLIM model (Second GenerationLouvain-la-Neuve Ice-ocean Model) which is produced in UCL. This model focuses on the sea-land continuity and solves equations related to geophysical, groundwater and environmental problems. The SLIM model uses an algorithm called the discontinuous Galerkin method. This method works pretty well for fluid simulations since it can fit quite easily any complex geometries oppositely to finite differences method. However as all the schemes with a fixed grid, this method has a lot of difficulties to manage the boundaries of the flow that changes throughout time. This is a considerable problem since a lot of applications are concerned by the tides, floods, storm surges, etc. In order to solve this issue, we will use wetting-drying methods. A multitude of them exists but in this thesis, we focus on a method inspired by an article written by A. Meister and S. Ortleb. This algorithm is interesting because it allows us to solve implicitly our problem with schemes with an order higher than one. Indeed, it is interesting because a lot of methods already solve explicitly this issue but it is not the case for implicit time integration. In this thesis, we will solve one-dimensional problem. We will explain the discontinuous Galerkin method and introduce an explicit wetting-drying method, which uses a thin layer of fluid throughout the domain. We will explain our method that combines the explicit algorithm, we mentioned, and the work done by A.Meister and S.Ortleb. We will test this method with well-known problems proposed by Balzano and Rittler. The results were conclusive for the problems proposed by Balzano. Indeed, our method was able to reproduce the physics correctly. However, the algorithm gives imprecise solution for the test case of Rittler. Thus, it would seem that the method lacks of precision. All in all, the algorithm, we proposed, seem to be a step in the right direction for solving implicitly the wetting-drying problem.