Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included within the article.

References

  1. Li SH, Wu LZ, Chen JJ, Huang RQ (2020) Multiple data-driven approach for predicting landslide deformation. Landslides 17:709–718

    Google Scholar 

  2. Jiang SH, Liu X, Huang J (2020) Non-intrusive reliability analysis of unsaturated embankment slopes accounting for spatial variabilities of soil hydraulic and shear strength parameters. Eng Comput. https://doi.org/10.1007/s00366-020-01108-6

    Article  Google Scholar 

  3. Li SH, Luo XH, Wu LZ (2021) A new method for calculating failure probability of landslide based on ANN and convex set model. Landslides 18:2855–2867

    Google Scholar 

  4. Wu LZ, Zhao DJ, Zhu JD, Peng JB, Zhou Y (2020) A Late Pleistocene river-damming landslide, Minjiang River, China. Landslides 17:433–444

    Google Scholar 

  5. Zhu SR, Wu LZ, Peng JB (2020) An improved Chebyshev semi-iterative method for simulating rainfall infiltration in unsaturated soils and its application to shallow landslides. J Hydrol 590:125157

    Google Scholar 

  6. Richards LA (1931) Capillary conduction of liquids through porous mediums. Physics 1(5):318–333

    MATH  Google Scholar 

  7. Gardner WR (1958) Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci 85(4):228–232

    Google Scholar 

  8. van Genuchten MTh (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils1. Soil Sci Soc Am J 44(5):892–898

    Google Scholar 

  9. Cooley RL (1983) Some new procedures for numerical solution of variably saturated flow problems. Water Resour Res 19(5):1271–1285

    Google Scholar 

  10. Wu LZ, Zhang LM, Huang RQ (2012) Analytical solution to 1D coupled water infiltration and deformation in two-layer unsaturated soils. Int J Numer Anal Meth Geomech 36(6):798–816

    Google Scholar 

  11. Zhu SR, Wu LZ, Huang J (2021) Application of an improved P(m)-SOR iteration method for flow in partially saturated soils. Comput Geosci. https://doi.org/10.1007/s10596-021-10114-6

    Article  MATH  Google Scholar 

  12. Wu LZ, Jinsong H, Fan W, Li X (2020) Hydro-mechanical coupling in unsaturated soils covering a non-deformable structure. Comput Geotech 117:103287

    Google Scholar 

  13. Srivastava R, Yeh TCJ (1991) Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous layered soils. Water Resour Res 27(5):753–762

    Google Scholar 

  14. Wu LZ, Liu GG, Wang LC, Zhang LM, Li BE, Li B (2016) Numerical analysis of 1D coupled infiltration and deformation in layered unsaturated porous medium. Environ Earth Sci 75(9):1–11

    Google Scholar 

  15. Zeng JC, Zha YY, Yang JZ (2018) Switching the Richards’ equation for modeling soil water movement under unfavorable conditions. J Hydrol 563:942–949

    Google Scholar 

  16. Zhu SR, Wu LZ, Shen ZH, Huang RQ (2019) An improved iteration method for the numerical solution of groundwater flow in unsaturated soils. Comput Geotech. 114:103113

    Google Scholar 

  17. Eini N, Afshar MH, Faraji Gargari S, Shobeyri G, Afshar A (2020) A fully Lagrangian mixed discrete least squares meshfree method for simulating the free surface flow problems. Eng Comput. https://doi.org/10.1007/s00366-020-01157-x

    Article  Google Scholar 

  18. Liu CY, Ku CY, Huang CC (2015) Numerical solutions for groundwater flow in unsaturated layered soil with extreme physical property contrasts. Int J Nonlinear Sci Numer Simul 16(7):325–335

    MathSciNet  MATH  Google Scholar 

  19. Crevoisier D, Chanzy A, Voltz M (2009) Evaluation of the Ross fast solution of Richards’ equation in unfavourable conditions for standard finite element methods. Adv Water Resour 32:936–947

    Google Scholar 

  20. Eymard R, Hilhorst D, Vohralik M (2006) A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer Math 105:73–131

    MathSciNet  MATH  Google Scholar 

  21. Pop IS, Schweizer B (2011) Regularization schemes for degenerate Richards equations and outflow conditions. Math Models Methods Appl Sci 21(8):1685–1712

    MathSciNet  MATH  Google Scholar 

  22. Ku CY, Liu CY, Su Y, Xiao JE (2018) Modeling of transient flow in unsaturated geomaterials for rainfall-induced landslides using a novel spacetime collocation method. Geofluids 2018:7892789

    Google Scholar 

  23. Dolejší V, Kuraz M, Solin P (2019) Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows. Appl Math Model 72:276–305

    MathSciNet  MATH  Google Scholar 

  24. Wu LZ, Zhu SR, Peng JB (2020) Application of the Chebyshev spectral method to the simulation of groundwater flow and rainfall-induced landslides. Appl Math Model 80:408–425

    MathSciNet  MATH  Google Scholar 

  25. Radu FA, Pop IS, Knabner P (2006) On the convergence of the Newton method for the mixed finite element discretization of a class of degenerate parabolic equation, Numerical Mathematics and Advanced Applications, pp. 1194–1200. Springer.

  26. Lehmann F, Ackerer P (1998) Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media. Transp Porous Media 31(3):275–292

    Google Scholar 

  27. List F, Radu FA (2016) A study on iterative methods for solving Richards’ equation. Comput Geosci 20(2):341–353

    MathSciNet  MATH  Google Scholar 

  28. Zha Y, Yang J, Yin L, Zhang Y, Zeng W, Shi L (2017) A modified Picard iteration scheme for overcoming numerical difficulties of simulating infiltration into dry soil. J Hydrol 551:56–69

    Google Scholar 

  29. Farthing MW, Ogden FL (2017) Numerical solution of Richards’ equation: a review of advances and challenges. Soil Sci Soc Am J 81(6):1257–1269

    Google Scholar 

  30. Zha Y, Yang J, Zeng J, Tso CM, Zeng W, Shi L (2019) Review of numerical solution of Richardson-Richards equation for variably saturated flow in soils. Wiley Interdisciplinary Reviews: Water: e1364.

  31. Hagemam LA, Young DM (1981) Applied Iterative methods. Academic Press, New York

    Google Scholar 

  32. Liu CS (2013) A two-side equilibration method to reduce the condition number of an ill-posed linear system. Comput Model Eng Sci 91(1):17–42

    MathSciNet  MATH  Google Scholar 

  33. Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869

    MathSciNet  MATH  Google Scholar 

  34. Dehghan M, Mohammadi-Arani R (2016) Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems. Comput Appl Math 36(4):1591–1606

    MathSciNet  MATH  Google Scholar 

  35. Bai ZZ, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24(3):603–626

    MathSciNet  MATH  Google Scholar 

  36. Bai ZZ, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87(3–4):93–111

    MathSciNet  MATH  Google Scholar 

  37. Dehghan M, Shirilord A (2019) Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems. Numer Algorithms 83(1):281–304

    MathSciNet  MATH  Google Scholar 

  38. Bai ZZ, Golub GH, Pan JY (2004) Preconditioned hermitian and skew-hermitian splitting methods for non-hermitian positive semidefinite linear systems. Numer Math 98(1):1–32

    MathSciNet  MATH  Google Scholar 

  39. Pop IS, Radu FA, Knabner P (2004) Mixed finite elements for the Richards’ equations: linearization procedure. J Comput Appl Math 168:365–373

    MathSciNet  MATH  Google Scholar 

  40. Lott PA, Walker HF, Woodward CS, Yang UM (2012) An accelerated Picard method for nonlinear systems related to variably saturated flow. Adv Water Resour 38:92–101

    Google Scholar 

  41. Mitra K, Pop IS (2019) A modified l-Scheme to solve nonlinear diffusion problems. Comput Math Appl 77:1722–1738

    MathSciNet  MATH  Google Scholar 

  42. Illiano D, Pop IS, Radu FA (2021) Iterative schemes for surfactant transport in porous media. Comput Geosci 25(2):805–822

    MathSciNet  MATH  Google Scholar 

  43. Solin P, Kuraz M (2011) Solving the nonstationary Richards equation with adaptive hp-FEM. Adv Water Resour 34(9):1062–1081

    Google Scholar 

  44. Liu W (2017) A two-grid method for the semi-linear reaction–diffusion system of the solutes in the groundwater flow by finite volume element. Math Comput Simul 142:34–50

    MathSciNet  MATH  Google Scholar 

  45. Chávez-Negrete C, Domínguez-Mota FJ, Santana-Quinteros D (2018) Numerical solution of Richards’ equation of water flow by generalized finite differences. Comput Geotech 101:168–175

    Google Scholar 

  46. Ku CY, Hong LD, Liu CY et al (2021) Space–time polyharmonic radial polynomial basis functions for modeling saturated and unsaturated flows. Eng Comput. https://doi.org/10.1007/s00366-021-01519-z

    Article  Google Scholar 

  47. Fahs M, Younes A, Lehmann F (2009) An easy and efficient combination of the mixed finite element method and the method of lines for the resolution of Richards’ equation. Environ Model Softw 24:1122–1126

    Google Scholar 

  48. Seus D, Mitra K, Pop IS, Radu FA, Rohde C (2018) A linear domain decomposition method for partially saturated flow in porous media. Comput Methods Appl Mech Eng 333:331–355

    MathSciNet  MATH  Google Scholar 

  49. Zhang Z, Wang W, Yeh TJ, Chen L, Wang Z, Duan L, An K, Gong C (2016) Finite analytic method based on mixed-form Richards’ equation for simulating water flow in vadose zone. J Hydrol 537:146–156

    Google Scholar 

  50. Benzi M (2002) Preconditioning techniques for large linear systems: a survey. J Comput Phys 182(2):418–477

    MathSciNet  MATH  Google Scholar 

  51. Das R (2017) GPUs in subsurface simulation: an investigation. Eng Comput 33(4):919–934

    MathSciNet  Google Scholar 

  52. Dehghan M, Hajarian M (2012) Modied AOR iterative methods to solve linear systems. J Vib Control 20(5):661–669

    MathSciNet  MATH  Google Scholar 

  53. Dehghan M, Dehghani-Madiseh M, Hajarian M (2013) A generalized preconditioned MHSS method for a class of complex symmetric linear systems. Math Model Anal 18(4):561–576

    MathSciNet  MATH  Google Scholar 

  54. Wang ZQ, Yin JF, Dou QY (2020) Preconditioned modified Hermitian and skew-Hermitian splitting iteration methods for fractional nonlinear Schrödinger equations. J Computat Appl Math 367:112420

    MATH  Google Scholar 

  55. Chao Z, Xie D, Sameh AH (2020) Preconditioners for nonsymmetric indefinite linear systems. J Comput Appl Math 367:112436

    MathSciNet  MATH  Google Scholar 

  56. Liu CY, Ku CY, Xiao JE, Huang CC, Hsu SM (2017) Numerical modeling of unsaturated layered soil for rainfall-induced shallow landslides. J Environ Eng Landsc Manag 25(4):329–341

    Google Scholar 

  57. Wang HF, Anderson MP (1982) Introduction to groundwater modeling. Freeman

    Google Scholar 

  58. Arnoldi WE (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q Appl Math 9(1):17–29

    MathSciNet  MATH  Google Scholar 

  59. Gentleman WM (1973) Least squares computations by givens transformations without square roots. IMA J Appl Math 12(3):329–336

    MathSciNet  MATH  Google Scholar 

  60. Tracy FT (2006) Clean two- and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers. Water Resour Res 42(8):1–11

    Google Scholar 

  61. Green WH, Ampt GA (1911) Studies on soil physics I. The flow of air and water through soils. J Agric Sci 4:1–24

    Google Scholar 

  62. Huyakorn PS, Springer EP, Guvanasen V, Wadsworth TD (1986) A three-dimensional finite-element model for simulating water flow in variably saturated porous media. Water Resour Res 22(13):1790–1808

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Key R&D Program of China [no. 2018YFC1504702]; and the National Natural Science Foundation of China [no. 41790432].

Author information

Authors and Affiliations

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

    S. R. Zhu & L. Z. Wu

  2. College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu, 610059, China

    S. R. Zhu

  3. State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031, China

    X. L. Song

Authors
  1. S. R. Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Z. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. L. Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to L. Z. Wu or X. L. Song.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, S.R., Wu, L.Z. & Song, X.L. An improved matrix split-iteration method for analyzing underground water flow. Engineering with Computers 39, 2049–2065 (2023). https://doi.org/10.1007/s00366-021-01551-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01551-z

Keywords

Search

Navigation