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A new mathematical model for traveling sand dunes: analysis and approximation. (English) Zbl 1445.76088

Summary: We present a new two-layer closed form model for the dynamics of desert dunes under the effect of a horizontal wind blowing in an arbitrary direction. This model is an extension of a very simplified model previously introduced by K. P. Hadeler and C. Kuttler [Granular matter. Int. Report Univ. Tübingen n. 185 (2003)]. Our extension, inspired by the sandpile dynamics approach, includes the effects of gravity on both sides (upwind and downwind) of the dune, and allows to describe erosion and deposition in a more accurate way. After a discussion of the model and its properties, we present a numerical scheme based on finite differences in 1D and we prove its consistency and stability. Some numerical tests show a good qualitative behavior and a realistic shape for the evolving dunes. Finally, we discuss the preliminary steps of a possible extension of this model to the 2D case.

MSC:

76T25 Granular flows
76M20 Finite difference methods applied to problems in fluid mechanics
86A60 Geological problems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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