This PhD thesis is devoted to the development and implementation of second-order well-balanced Lagrange-projection numerical methods for hyperbolic partial differential equations. In particular, the Lagrange-projection formalism entails a decomposition of the acoustic and transport terms of the model, while the well-balanced property represents the ability of the scheme of preserving the stationary solutions of the model.

Here we mainly focus on the numerical approximation of the shallow-water system coupled with the Exner equation, where the latter expresses the evolution in time of the bed elevation. It is known that it is not a trivial task to numerically simulate the resulting shallow-water-Exner model, as a decoupled method could lead to the presence of spurious oscillations in the numerical outputs. In addition, when considering the Lagrange-projection formalism, while it is clear how to decompose the shallow water system, this is not true when it comes to the Exner equation. For this reason we investigate different possible numerical strategies.