A multi level linearized Crank-Nicolson scheme for Richards equation under variable flux boundary conditions

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 rain attacks. Efficient and precise linearized numerical schemes are necessary, but there is few study related it, and the numerical theory is incomplete because of the degeneracy and strong nonlinearity. In this paper, we establish a linearized Crank-Nicolson finite difference scheme which is a three-level scheme with almost second-order accuracy. In stability analysis, we develop a creative technique to overcome the degeneracy by adding a small positive perturbation epsilon. We also propose the error estimates by applying Young's inequality and prove the convergence order is approximate to second-order. Numerical examples are also provided to verify our main results and show the relationship between the computational error and epsilon is linear.