关于多孔介质中输运与扩散竞争水流数学模型的一些注记
Some remarks on a mathematical model for water flow in porous media with competition between transport and diffusion
AI总结：
本文基于Detmann等人的模型，用非线性偏微分方程描述多孔介质水流中输运与扩散的竞争，采用时间离散化和有限元方法进行数值求解，并展示模拟结果。
摘要：
本文涉及多孔介质中流体流动的数学建模，特别是土壤中的水流，旨在描述输运与扩散之间的竞争。分析基于B. Detmann、C. Gavioli和P. Krejčí开发的数学模型，该模型以新颖的方式包含了重力效应。该模型由一个非线性偏微分方程组成，描述了水的扩散和重力输运。尽管在某些特殊情况下可以获得解析解，但在更一般的情况下仅有数值解可用。求解算法基于时间离散化和有限元方法，并用Matlab编写。文中展示了数值模拟的结果，并讨论了模型的行为。
英文摘要：
The contribution deals with the mathematical modelling of fluid flow in porous media, in particular water flow in soils, with the aim of describing the competition between transport and diffusion. The analysis is based on a mathematical model developed by B. Detmann, C. Gavioli, and P. Krejčí, in which the effects of gravity are included in a novel way. The model consists of a nonlinear partial differential equation describing both the diffusion and the gravitational transport of water. Although analytical solutions can be obtained for some special cases, only numerical solutions are available in more general situations. The solving algorithm is based on a time discretisation and the finite element method, and is written in Matlab. The results of the numerical simulations are shown and the behaviour of the model is discussed.
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