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

ABSTRACT

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 ϵ. 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 ϵ is linear.

KEYWORDS:

• Richards equation
• finite-difference scheme
• stability
• error estimate

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K59
• 65M06
• 65M12
• 65M15
• 76S05

Acknowledgments

The draft of this paper was finished when FZ 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.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

FL was supported by the Fundamental Research Funds for the Central Universities (grant no. DUT17 RC(3)053); 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).