Modeling Variably Saturated Flow Problems Using Newton-Type Linearization Methods

  • C. Paniconi
  • M. Putti
Part of the International Centre for Mechanical Sciences book series (CISM, volume 364)


Numerical procedures to solve the nonlinear equation governing flow in variably saturated porous media commonly involve Newton or Picard iteration. The former scheme is stable and quadratically convergent in a local sense, but costly and algebraically complex. The latter scheme is simple and cheap, but slower converging and not as robust. We present a common framework for comparing these two methods, and introduce other approaches that range from simplifications of the Picard scheme to approximations of Newton’s method. These other approaches include explicit discretizations, first and second order accurate linearizations, and quasi-Newton schemes. Relaxation and line search algorithms to accelerate convergence of the Picard, Newton, and quasi-Newton methods will also be considered. The effectiveness of these various iterative and noniterative methods will be assessed according to criteria of efficiency and robustness.


Newton Method Line Search Pressure Head Time Step Size Saturated Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Philip, J. R., Theory of infiltration, Adv. Hydrosci. 5, 215–296, 1969.CrossRefGoogle Scholar
  2. 2.
    Ames, W. F., Numerical Methods for Partial Differential Equations. Academic Press, San Diego, CA second edition, 1977.Google Scholar
  3. 3.
    Huyakorn, P. S. and G. F. Pinder, Computational Methods in Subsurface Flow. Academic Press, New York, NY, 1983.MATHGoogle Scholar
  4. 4.
    Paniconi, C. and E. F. Wood, A detailed model for simulation of catchment scale subsurface hydrologic processes, Water Resour. Res. 29 (6), 1601–1620, 1993.CrossRefGoogle Scholar
  5. 5.
    Cooley, R. L., Some new procedures for numerical solution of variably saturated flow problems, Water Resour. Res. 19 (5), 1271–1285, 1983.CrossRefGoogle Scholar
  6. 6.
    Huyakorn, P. S., E. P. Springer, V. Guvanasen and T. D. Wadsworth, A three-dimensional finite-element model for simulating water flow in variably saturated porous media, Water Resour. Res. 22 (13), 1790–1808, 1986.CrossRefGoogle Scholar
  7. 7.
    Paniconi, C. and M. Putti, A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems, Water Resour. Res. 30 (12), 3357–3374, 1994.CrossRefGoogle Scholar
  8. 8.
    Axelsson, O., Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations, Linear Algebra Appl. 29, 1–16, 1980.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pini, G., G. Gambolati and G. Galeati, 3-D finite element transport models by upwind preconditioned conjugate gradients, Adv. Water Resour. 12, 54–58, 1989.CrossRefGoogle Scholar
  10. 10.
    van der Vorst, H., Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of rionsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13, 631–644, 1992.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Freund, R. W., A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput. 14 (2), 470–482, 1993.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kershaw, D. S., The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations, J. Comput. Phys. 26, 43–65, 1978.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gambolati, G. and A. Perdon, Conjugate gradients in subsurface flow and land subsidence modeling. In: Bear, J. and M. Y. Corapcioglu (eds.) Fundamentals of Transport Phenomena in Porous Media. Martinus Nijhoff, Dordrecht, Holland, pp 953–984, 1984.Google Scholar
  14. 14.
    Pullan, A. J., The quasilinear approximation for unsaturated porous media flow, Water Resour. Res. 26 (6), 1219–1234, 1990.CrossRefGoogle Scholar
  15. 15.
    van Genuchten, M. T. and D. R. Nielsen, On describing and predicting the hydraulic properties of unsaturated soils, Ann. Geophys. 3 (5), 615–628, 1985.Google Scholar
  16. 16.
    Paniconi, C., A. A. Aldama and E. F. Wood, Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear Richards equation, Water Resour. Res. 27 (6), 1147–1163, 1991.CrossRefGoogle Scholar
  17. 17.
    Huyakorn, P. S., S. D. Thomas and B. M. Thompson, Techniques for making finite elements competitive in modeling flow in variably saturated porous media, Water Resour. Res. 20 (8), 1099–1115, 1984.CrossRefGoogle Scholar
  18. 18.
    Ortega, J. M. and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, NY, 1970.MATHGoogle Scholar
  19. 19.
    Stoer, J. and R. Bulirsch, Introduction to Numerical Analysis. Springer-Verlag, New York, NY, 1980.Google Scholar
  20. 20.
    Lees, M., A linear three-level difference scheme for quasilinear parabolic equations, Math. Comp. 20, 516–522, 1966.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Dennis, J. E. and J. J. Moré, Quasi-Newton methods, motivation and theory, SIAM Review 19 (1), 46–89, 1977.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Matthies, H. and G. Strang, The solution of nonlinear finite element equations, Int.1. Numer. Meth. Eng. 14, 1613–1626, 1979.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Geradin, M., S. Idelsohn and M. Hogge, Computational strategies for the solution of large nonlinear problems via quasi-Newton methods, Computers 6 Structures 13, 73–81, 1981.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Putti, M. and C. Paniconi, Quasi-Newton methods for Richards’ equation. Irr: Peters, A., G. Wittum, B. Herrling, U. Meissner, C. A. Brebbia, W. G. Gray and G. F. Pinder (eds.) Computational Methods in Water Resources X, Volume 1. Kluwer Academic, Dordrecht, Holland, pp 99–106, 1994.Google Scholar
  25. 25.
    Paniconi, C. and M. Putti, Quasi-Newton and line search methods for the finite element solution of unsaturated flow problems. In: Wang, S. S. Y. (ed.) Second International Conference on Hydro-Science and Engineering, Beijing, China, 1995.Google Scholar
  26. 26.
    Papadrakakis, M., Solving large-scale nonlinear problems in solid and structural mechanics. In: Papadrakakis, M. (ed.) Solving Large-Scale Problems in Mechanics 183–223, John Wiley and Sons, New York, NY, 1993.Google Scholar
  27. 27.
    Dennis, J. E. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ, 1983.MATHGoogle Scholar
  28. 28.
    Fletcher, R., Practical Methods of Optimization, Vol. 1: Unconstrained Optimization. John Wiley and Sons,-New York, NY, 1980.Google Scholar
  29. 29.
    Putti, M. and C. Paniconi, Evaluation of the Picard and Newton iteration schemes for three-dimensional unsaturated flow. In: Russell, T. F., R. E. Ewing, C. A. Brebbia, W. G. Gray and G. F. Pinder (eds.) Proceedings of the IX International Conference on Computational Methods in Water Resources, Vol. 1, Numerical Methods in Water Re-sources. Computational Mechanics Publications, Southampton, UK, pp 529–536, 1992.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • C. Paniconi
    • 1
  • M. Putti
    • 2
  1. 1.CRS4CagliariItaly
  2. 2.University of PaduaPaduaItaly

Personalised recommendations