A Linearized Finite Difference Scheme for the Richards Equation Under Variable-Flux Boundary Conditions

被引:0
作者
Liu Fengnan
Yasuhide Fukumoto
Xiaopeng Zhao
机构
[1] Dalian University of Technology,School of Mathematics and Physics Science
[2] Kyushu University,Institute of Mathematics for Industry
[3] Northeastern University,College of Sciences
来源
Journal of Scientific Computing | 2020年 / 83卷
关键词
Richards equation; Finite difference scheme; Stability; Error estimate; 35K59; 65M06; 65M12; 65M15; 76S05;
D O I
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中图分类号
学科分类号
摘要
The Richards equation is a degenerate nonlinear PDE that models a flow through saturated/unsaturated porous media. Research on its numerical methods has been conducted in many fields. Implicit schemes based on a backward Euler format are widely used in calculating it. However, it is difficult to obtain stability with a numerical scheme because of the strong nonlinearity and degeneracy. In this paper, we establish a linearized semi-implicit finite difference scheme that is faster than backward Euler implicit schemes. We analyze the stability of this scheme by adding a small positive perturbation ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} to the coefficient function of the Richards equation. Moreover, we show that there is a linear relationship between the discretization error in the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}-norm and ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}. Numerical experiments are carried out to verify our main results.
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