Application of improved Picard method in unsaturated seepage

Richards' equation is widely used in the simulation of unsaturated seepage and related fields. In the numerical solution, the finite volume method can be used for numerical discretization, and then the Picard method can be used for iterative solution. However, in order to obtain reliable and accurate numerical solutions, the spatial step size of a uniform grid is usually very small, especially under some unfavorable numerical conditions, such as rainfall infiltration into dry soil, which often makes the iterative process time-consuming or even unable to converge. Therefore, combined with the non-uniform grid in the form of Chebyshev, an improved Picard iteration method (NTG-PI) is proposed based on the non-uniform two-grid correction scheme. Through three examples of unsaturated seepage, the numerical accuracy, convergence rate and acceleration effect of the improved algorithm are validated by comparing the traditional methods and analytical solutions. The results show that compared with the traditional Picard and the adaptive relaxation Picard methods, the proposed method NTG-PI can obtain higher numerical accuracy with a smaller number of discrete nodes, and also has higher computational efficiency. The proposed method can provide a reference for the numerical simulation of unsaturated seepage.