Résumé
This work is devoted to the development of numerical schemes to approximatesolutions of depth averaged flow models.We first detail the construction of Finite Volume approaches for the ShallowWater system with source terms on unstructured meshes. Based on a suitablereformulation of the equations, we implement a well-balanced and positivepreservingapproach, and suggest adapted MUSCL extensions. The methodis shown to handle irregular topography variations and demonstrates strongstabilities properties. The inclusion of friction terms is subject to a thoroughanalysis, leading to the establishment of some Asymptotic Preserving propertythrough the enhancement of another recent Finite Volume scheme.The second aspect of this study concerns discontinuous Galerkin Finite-Element methods. Some of the ideas advanced in the Finite Volume context areemployed to broach the Shallow Water system on triangular meshes. Numericalresults are exposed and the method turns out to be well suited to describe a largevariety of flows. On these observations we finally propose to exploit its featuresto extend the approach to a new family of Green-Nadghi equations. Numericalexperiments are also proposed to validate this numerical model.