加速型改进迭代法在非饱和土渗流中的应用研究

Application research on accelerated modified iteration method in unsaturated flow

Richards方程常用于非饱和土渗流问题,并且应用广泛。在数值求解中,对Richards方程线性化,进而采用有限差分法进行数值离散以及迭代计算。其中传统的迭代法比如Jacobi迭代、Gauss-Seidel迭代法(GS)和连续超松驰迭代法(successive over-relaxation method,简称SOR)迭代收敛率较慢,尤其在离散空间步长较小以及离散时间步长较大时。因此,采用整体校正法以及多步预处理法对传统迭代法进行改进,提出一种基于整体校正法的多步预处理Gauss-Seidel迭代法(improved Gauss-Seidel iterative method with multistep preconditioner based on the integral correction method,简称ICMP(m)-GS)求解Richards方程导出的线性方程组。通过非饱和渗流算例,并与传统迭代法和解析解对比,对改进算法的收敛率和加速效果进行了验证。结果表明,提出的ICMP(m)-GS可以很大程度地改善线性方程组的病态性,相较于常规方法GS,SOR以及单一改进方法,ICMP(m)-GS具有更快的收敛率,更高的计算效率和计算精度。该方法可以为非饱和土渗流的数值模拟提供一定参考。

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.