Modelling unsaturated flow in porous media using an improved iterative scheme

Richards’ equation is often used in unsaturated flow problems and has a wide range of applications. In the numerical solution, the Richards’ equation is linearized first, and then the finite difference method is used for numerical discretization and iterative calculation. The traditional iterative methods such as Jacobi, Gauss–Seidel (GS) and SOR iterative methods have a slower convergence rate, especially when the discrete space step is small and the time step is large. Therefore, we adopt the integral correction method and the multistep preconditioner to improve the traditional iterative methods and propose an improved Gauss–Seidel iterative method (ICMP(m)-GS) with multistep preconditioner based on the integral correction method to solve the linear equations derived from linearized Richards’ equation. Through examples of unsaturated flow, convergence rate and computational accuracy of the proposed algorithm are validated by comparing the traditional methods and analytical solutions. The results show that the proposed ICMP(m)-GS can greatly improve the ill-condition of linear equations. Compared with the conventional methods GS, SOR and a single improvement method, ICMP(m)-GS has a faster convergence rate, higher calculation efficiency and calculation accuracy. The application example shows that the proposed method can also well simulate the time-varying law of pressure head in the rainfall infiltration of unsaturated soil slopes, and has a nice application effect.