Abstract 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 Hadeler and Kuttler [12] . 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.

中文翻译：

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一种新的移动沙丘数学模型：分析和近似

摘要 我们提出了一种新的两层封闭形式模型，用于在任意方向的水平风的影响下沙漠沙丘的动力学。该模型是先前由 Hadeler 和 Kuttler [12] 引入的非常简化的模型的扩展。我们的扩展受到沙堆动力学方法的启发，包括沙丘两侧（上风和下风）的重力影响，并允许以更准确的方式描述侵蚀和沉积。在讨论了模型及其属性之后，我们提出了一个基于一维有限差分的数值方案，并证明了它的一致性和稳定性。一些数值测试表明，演化沙丘具有良好的定性行为和逼真的形状。最后，我们讨论了将该模型可能扩展到 2D 案例的初步步骤。