One-Dimensional Finite Element Method Solution of a Class of Integro-Differential Equations: Application to Non-Fickian Transport in Disordered Media

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• 作者：Rami Ben-Zvi ; Harvey Scher ; Shidong Jiang ; Brian Berkowitz
• 关键词：Continuous time random walk ; Heterogeneous porous media ; Prony model
• 刊名：Transport in Porous Media
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：115
• 期：2
• 页码：239-263
• 全文大小：922 KB
• 刊物类别：Earth and Environmental Science
• 刊物主题：Earth sciences
  Geotechnical Engineering
  Industrial Chemistry and Chemical Engineering
  Civil Engineering
  Hydrogeology
  Mechanics, Fluids and Thermodynamics
  
• 出版者：Springer Netherlands
• ISSN：1573-1634
• 卷排序：115

文摘

We study an integro-differential equation that has important applications to problems of anomalous transport in highly disordered media. In one application, the equation is the continuum limit of a continuous time random walk used to quantify non-Fickian (anomalous) contaminant transport. The finite element method is used for the spatial discretization of this equation, with an implicit scheme for its time discretization. To avoid storage of the entire history, an efficient sum-of-exponential approximation of the kernel function is constructed that allows a simple recurrence relation. A 1D formulation with a linear element is implemented to demonstrate this approach, by comparison with available experiments and with an exact solution in the Laplace domain, transformed numerically to the time domain. The proposed scheme convergence assessment is briefly addressed. Future extensions of this implementation are then outlined.