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Newton-Type Linearization and Line Search Methods for Unsaturated Flow Models

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Advances in Groundwater Pollution Control and Remediation

Part of the book series: NATO ASI Series ((ASEN2,volume 9))

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Abstract

In this paper we describe various iterative and noniterative approaches for solving the nonlinear equation governing flow in variably saturated porous media. The approaches are described within the framework of the classical Newton-Raphson method, and include simplified, approximate, combined, and accelerated methods. We give results of numerical tests conducted for one and two-dimensional problems involving both steady state and transient flow.

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References

  • Ames WF (1977) Numerical Methods for Partial Differential Equations. Academic Press, San Diego, CA second edition

    Google Scholar 

  • Axelsson 0 (1980) Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations. Linear Algebra Appl. 29:1–16

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Dennis JE, Moré J J (1977) Quasi-Newton methods, motivation and theory. SIAM Review 19(l):46–89

    Article  Google Scholar 

  • Dennis JE, Schnabel RB (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ

    Google Scholar 

  • Fletcher R (1980)Practical Methods of Optimization, Vol. 1: Unconstrained Optimization. John Wiley & Sons, New York, NY

    Google Scholar 

  • Freund RW (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14(2):470–482

    Article  Google Scholar 

  • Gambolati G, Perdon A (1984) Conjugate gradients in subsurface flow and land subsidence modeling. In: Bear J, Corapcioglu MY (eds) Fundamentals of Transport Phenomena in Porous Media Martinus Nijhoff, Dordrecht, Holland, pp 953–984

    Google Scholar 

  • Geradin M, Idelsohn S, Hogge M (1981) Computational strategies for the solution of large nonlinear problems via quasi-Newton methods. Computers Structures 13:73-81

    Article  Google Scholar 

  • Huyakorn PS, Pinder GF (1983) Computational Methods in Subsurface Flow. Academic Press, NewYork, NY

    Google Scholar 

  • Huyakorn PS, Thomas SD, Thompson BM (1984) Techniques for making finite elements competitive in modeling flow in variably saturated porous media. Water Resour Res. 20(8):1099—1115

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Kershaw DS (1978) The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations. J. Comput. Phys. 26:43–65

    Article  Google Scholar 

  • Lees M (1966) A linear three-level difference scheme for quasilinear parabolic equations. Math. Comp. 20:516–522

    Article  Google Scholar 

  • Matthies H, Strang G (1979) The solution of nonlinear finite element equations. Int. J. Numer. Meth. Eng. 14:1613–1626

    Article  Google Scholar 

  • Ortega JM, Rheinboldt WC (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, NY

    Google Scholar 

  • Paniconi C, Aldama A A, Wood EF (1991) Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear Richards equation. Water Resour. Res. 27(6):1147—1163

    Article  Google Scholar 

  • Paniconi C, Wood EF (1993) A detailed model for simulation of catchment scale subsurface hydrologic processes. Water Resour. Res. 29(6):1601—1620

    Article  Google Scholar 

  • Paniconi C, Putti M (1994) A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems. Water Resour. Res. 30(12):3357—3374

    Article  Google Scholar 

  • Paniconi C, Putti M (1995) Quasi-Newton and line search methods for the finite element solution of unsaturated flow problems. In: Advances in Hydro-Science and -Engineering, Volume II, Part B Tsinghua University Press, Beijing, China, pp 1681— 1688

    Google Scholar 

  • Papadrakakis M (1993) Solving large-scale nonlinear problems in solid and structural mechanics. In: Papadrakakis. M (ed) Solving Large-Scale Problems in Mechanics 183-223. John Wiley Sons New York, NY

    Google Scholar 

  • Philip JR (1969) Theory of infiltration. Adv. Hydrosci. 5:215–296

    Google Scholar 

  • Pini G, Gambolati G, Galeati G (1989) 3-D finite element transport models by upwind preconditioned conjugate gradients. Adv. Water Resour. 12:54–58

    Article  Google Scholar 

  • Pullan A J (1990) The quasilinear approximation for unsaturated porous media flow. Water Resour. Res. 26(6):1219–1234

    Article  Google Scholar 

  • Putti M, Paniconi C (1994) Quasi-Newton methods for Richards’ equation. In: Peters A, Wittum G, Herrling B, Meissner U, Brebbia CA, Gray WG, Pinder GF (eds) Computational Methods in Water Resources X, Volume 1 Kluwer Academic, Dordrecht, Holland, pp 99–106

    Google Scholar 

  • Stoer J, Bulirsch R (1980) Introduction to Numerical Analysis. Springer-Verlag, New York, NY

    Google Scholar 

  • van der Vorst H (1992) Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13:631–644

    Article  Google Scholar 

  • van Genuchten MT, Nielsen DR (1985) On describing and predicting the hydraulic properties of unsaturated soils. Ann. Geophys. 3(5):615–628

    Google Scholar 

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Author information

Authors and Affiliations

  1. CRS4, Via Nazario Sauro 10, 09123, Cagliari, Italy

    Claudio Paniconi

  2. Dipartimento di Metodi e Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131, Padova, Italy

    Mario Putti

Authors
  1. Claudio Paniconi
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  2. Mario Putti
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Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, School of Civil and Environmental Engineering, Atlanta, Georgia, USA

    Mustafa M. Aral

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© 1996 Kluwer Academic Publishers

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Paniconi, C., Putti, M. (1996). Newton-Type Linearization and Line Search Methods for Unsaturated Flow Models. In: Aral, M.M. (eds) Advances in Groundwater Pollution Control and Remediation. NATO ASI Series, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0205-3_7

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  • DOI: https://doi.org/10.1007/978-94-009-0205-3_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6576-4

  • Online ISBN: 978-94-009-0205-3

  • eBook Packages: Springer Book Archive

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