Modeling Variably Saturated Flow Problems Using Newton-Type Linearization Methods
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.
KeywordsNewton Method Line Search Pressure Head Time Step Size Saturated Flow
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- 2.Ames, W. F., Numerical Methods for Partial Differential Equations. Academic Press, San Diego, CA second edition, 1977.Google Scholar
- 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
- 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
- 19.Stoer, J. and R. Bulirsch, Introduction to Numerical Analysis. Springer-Verlag, New York, NY, 1980.Google Scholar
- 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.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.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
- 28.Fletcher, R., Practical Methods of Optimization, Vol. 1: Unconstrained Optimization. John Wiley and Sons,-New York, NY, 1980.Google Scholar
- 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