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The Richards equation is a nonlinear degenerate advection diffusion equation that models flow in saturated/unsaturated porous media, it's crucially important for prediction of disasters when heavy ...

The Richards equation is a degenerate nonlinear partial differential equation which serves as a model for describing a flow of water through saturated/unsaturated porous medium under the action of gravity. This paper develops a numerical method, with a mathematical support, for the one-dimensional Richards equation. Implicit schemes based on a backward Euler format have been widely used, but have a difficulty in insuring the stability, because of the strong nonlinearity and degeneracy. A linearized semi-implicit finite difference scheme that is faster than the backward Euler implicit schemes is established, the stability of this scheme is proved by adding a small perturbation to the coefficient function, and an error estimate is made. It is found that there is a linear relationship between the discretization error in a certain norm and the perturbation strength.

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.