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

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Abstract

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 \(\epsilon \) to the coefficient function of the Richards equation. Moreover, we show that there is a linear relationship between the discretization error in the \(L^{\infty }\)-norm and \(\epsilon \). Numerical experiments are carried out to verify our main results.

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Acknowledgements

LF was supported by Fundamental Research Funds for the Central Universities (No. DUT19RC(4)038); YF was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Grant No. 16K05476); XZ was supported by the China Postdoctoral Science Foundation (Grant No. 2015M581689). The draft of this paper was finished when LF and XZ visited the Institute of Mathematics for Industry of Kyushu University in the summer of 2018. They wish to appreciate the hospitality of Ms. Sasaguri.

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Authors and Affiliations

  1. School of Mathematics and Physics Science, Dalian University of Technology, Panjin, 124221, China

    Liu Fengnan

  2. Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Yasuhide Fukumoto

  3. College of Sciences, Northeastern University, Shenyang, 110819, China

    Xiaopeng Zhao

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  1. Liu Fengnan
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  2. Yasuhide Fukumoto
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  3. Xiaopeng Zhao
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Correspondence to Xiaopeng Zhao.

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Fengnan, L., Fukumoto, Y. & Zhao, X. A Linearized Finite Difference Scheme for the Richards Equation Under Variable-Flux Boundary Conditions. J Sci Comput 83, 16 (2020). https://doi.org/10.1007/s10915-020-01196-y

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  • DOI: https://doi.org/10.1007/s10915-020-01196-y

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