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Application of an improved P(m)-SOR iteration method for flow in partially saturated soils

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This paper studies the potential of using the successive over-relaxation iteration method with polynomial preconditioner (P(m)-SOR) to solve variably saturated flow problems described by the linearized Richards’ equation. The finite difference method is employed to numerically discretize and produce a system of linear equations. Generally, the traditional Picard method needs to re-evaluate the iterative matrix in each iteration, so it is time-consuming. And under unfavorable conditions such as infiltration into extremely dry soil, the Picard method suffers from numerical non-convergence. For linear iterative methods, the traditional Gauss-Seidel iteration method (GS) has a slow convergence rate, and it is difficult to determine the optimum value of the relaxation factor w in the successive over-relaxation iteration method (SOR). Thus, the approximate optimum value of w is obtained based on the minimum spectral radius of the iterative matrix, and the P(m)-SOR method is extended to model underground water flow in unsaturated soils. The improved method is verified using three test examples. Compared with conventional Picard iteration, GS and SOR methods, numerical results demonstrate that the P(m)-SOR has faster convergence rate, less computation cost, and good error stability. Besides, the results reveal that the convergence rate of the P(m)-SOR method is positively correlated with the parameter m. This method can serve as a reference for numerical simulation of unsaturated flow.

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Acknowledgements

We thank the Project supported by the National Key R&D Program of China (2018YFC1504702) and the National Natural Science Foundation of China (No. 41790432). The corresponding author thanks the Innovation Team of the Chengdu University of Technology.

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Authors and Affiliations

  1. State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, 400074, Chongqing, People’s Republic of China

    S. R. Zhu & L. Z. Wu

  2. College of Environment and Civil Engineering, Chengdu University of Technology, 610059, Chengdu, Sichuan, People’s Republic of China

    S. R. Zhu

  3. Discipline of Civil, Surveying and Environmental Engineering, Priority Research Centre for Geotechnical Science and Engineering, The University of Newcastle, Callaghan, NSW, Australia

    J. Huang

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Correspondence to L. Z. Wu.

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Zhu, S.R., Wu, L.Z. & Huang, J. Application of an improved P(m)-SOR iteration method for flow in partially saturated soils. Comput Geosci 26, 131–145 (2022). https://doi.org/10.1007/s10596-021-10114-6

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