An improved matrix split-iteration method for analyzing underground water flow

The Hermitian and skew-Hermitian splitting iteration method (HSS) is commonly an effective linear iterative method for solving sparse non-Hermite positive definite equations. However, it is time-consuming to solve linear equations. Hence, inexact Hermitian and skew-Hermitian splitting iteration approaches with multistep preconditioner (PIHSS(m)) are proposed for analyzing underground water flow. For unsaturated porous media, an exponential model is adopted to linearize the Richards equation. The governing equations are discretized using the finite element method to produce a system of linear equations. Furthermore, the inexact Hermitian and skew-Hermitian splitting iteration methods (IHSS) and PIHSS(m) are used to solve the linear equations. The results show that PIHSS(m) can effectively solve the 1D unsaturated flow problem and 2D transient drainage problem in partially and completely saturated soils. The IHSS has higher numerical accuracy than the classical methods such as Picard method and Gauss-Seidel iterative method. Compared with IHSS, PIHSS(m) achieves faster convergence rate and higher computational efficiency, particularly for solving groundwater flow problems with high grid density. Additionally, the numerical results reveal that PIHSS(m) has excellent acceleration, that is, at least 50% acceleration compared with the IHSS.

Automated summary

This article proposes an inexact Hermitian and skew-Hermitian splitting iteration method with multistep preconditioner for analyzing underground water flow. The results show that this method can effectively solve unsaturated flow and transient drainage problems, with higher numerical accuracy, faster convergence rate, and higher computational efficiency compared to classical methods.