Finite Element Method for Solving the Two-Dimensional Richards Equation in MATLAB

The core of this framework is to transform the nonlinear partial differential equation into a system of nonlinear algebraic equations through mathematical approaches including Green’s theorem, Jacobian transformation, and Gaussian numerical integration. The Newton-Raphson method is then adopted, combined with the diagonal normalization method for imposing boundary conditions, to finally accomplish the solution of the nonlinear algebraic system.

The numerical simulation code developed based on this framework has been effectively validated under two typical irrigation scenarios: subsurface pipe infiltration irrigation and surface drip irrigation. In both scenarios, the positions of the simulated wetting fronts were in good agreement with the reference results, verifying the accuracy and applicability of the model. The programming framework and code provided in this paper can help researchers with weak mathematical foundations quickly get started with finite element numerical simulation of the two-dimensional Richards equation, and are expected to play a positive role in promoting future breakthroughs in the numerical simulation of soil physical processes in China.