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

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.