Analytical and Numerical Study of a Model of Erosion and Sedimentation

We consider the following problem, arising within a geological model of sedimentationerosion: For a given vector field g and a given nonnegative function F defined on a oneor twodimensional domain Ω, find a vector field under the form g̃ = ug, with 0 ≤ u(x) ≤ 1 for a.e. x ∈ Ω, such that divg̃ + F ≥ 0 and (u − 1)(divg̃ + F ) = 0 in Ω. We first give a weak formulation of this problem, and we prove a comparison principle on a weak solution of the problem. Thanks to this property, we get the proof of the uniqueness of the weak solution. The existence of a solution results from the proof of the convergence of an original scheme. Numerical examples show the efficiency of this scheme and illustrate its convergence properties.