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.

Richards方程是一个退化的非线性PDE，用于模拟通过饱和/不饱和多孔介质的流动。在许多领域已经对其数值方法进行了研究。基于反向欧拉格式的隐式方案被广泛地用于计算它。但是，由于强烈的非线性和简并性，很难用数值方案获得稳定性。在本文中，我们建立了一个线性的半隐式有限差分方案，该方案比后向Euler隐式方案要快。我们通过向Richards方程的系数函数添加一个小的正摄动\（\ epsilon \）来分析该方案的稳定性。此外，我们证明\（L ^ {\ infty} \）-范数中的离散化误差与\（\ epsilon \）。进行数值实验以验证我们的主要结果。