关键词: Richards方程, Gauss-Seidel迭代, 收敛率, 整体校正, 多步预处理

Abstract: 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 iteration, Gauss-Seidel iteration (GS) and SOR iteration have a slower convergence rate, especially when the discrete space step size is small and the discrete time step size is large. Therefore, we adopt the integral correction method and the multistep preconditioner to improve the traditional iterative methods, and propose a improved Gauss-Seidel iterative method with multistep preconditioner based on the integral correction method (ICMP(m)-GS) to solve the linear equations derived from Richards equation. Through examples of unsaturated seepage flow, convergence rate and acceleration effect 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 faster convergence rate, higher calculation efficiency and calculation accuracy. This method can serve as a reference for numerical simulation of unsaturated flow.

Key words: Richards’ equation, Gauss-Seidel iteration, convergence rate, integral correction, multistep preconditioner