Analytical and Numerical Study of a Model of Erosion and Sedimentation

• Published in: SIAM journal on numerical analysis Vol. 43; no. 6; pp. 2344 - 2370
• Main Authors: Eymard, Robert, Gallouët, Thierry
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01-01-2006
• Subjects: Basins; Erosion; Exact sciences and technology; Geology; Hypotheses; Inverse problems; Mathematical analysis; Mathematical functions; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical schemes; Partial differential equations; Partial differential equations, boundary value problems; Perceptron convergence procedure; Scalars; Sciences and techniques of general use; Sedimentation & deposition; Sediments; Soil erosion; Uniqueness; Volume; erosion and sedimentation models; 65N12; Existence of solution; Erosion and sedimentation models; Process solution; Finite volume method; finite volume methods; Convergence; Numerical analysis; Weak solution; Hyperbolic inequality; Doubling variable technique; 35R45; process solutions; Weak uniqueness; hyperbolic inequalities; Vector field; 65N30
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/040605874

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 one-or twodimensional domain Ω, find a vector field under the form g̃ = ug, with 0 ≤ u(x) ≤ 1 for a.e. x ϵ 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.