Advertisement

Projection Methods for the Finite Element Solution of the Dual-Porosity Model in Variably Saturated Porous Media

  • Giuseppe Gambolati
  • Mario Putti
  • Claudio Paniconi
Part of the NATO ASI Series book series (ASEN2, volume 9)

Abstract

Projection methods based on Krylov subspaces are becoming increasingly popular for the solution of large sparse sets of nonsymmetric linear and nonlinear equations arising from the numerical integration of (initial) boundary value problems. One such problem is the so-called “two-site” or “dual-porosity” model describing the transport of reactive contaminants through a sorbing porous medium characterized by intra-aggregate diffusion. Finite element integration of the dual-porosity model yields large sparse nonsymmetric systems whose solution represents the largest fraction of the computational cost of simulation.

Keywords

Projection Method Krylov Subspace Couple Approach Krylov Subspace Method Minimal Residual 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnoldi WE (1951) The principle of minimized iteration in the solution of the matrix eigenvalue problem. Quart. ppl. Math. 9:17–29.Google Scholar
  2. Barret R, Berry M, Chan T, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, van der Vorst H (1994) Templates for the Solution of linear Systems: Building Blocks for Iterative Methods. S1AM publisher, Philadelphia, PA.Google Scholar
  3. Brusseau ML (1994) Transport of reactive contaminants in heterogeneous porous media. Rev. Geophys. 32(3):285–313.CrossRefGoogle Scholar
  4. Brusseau ML, Rao PSC (1989) Sorption nonideality during organic contaminant transport in porous media, Crit. Rev. Env. Contr. 19(1):33–99.CrossRefGoogle Scholar
  5. Concus P, Golub GH, O’Leary DP (1976) A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In: Bunch J, Rose D (eds) Sparse Matrix Computation Academic Press, New York.Google Scholar
  6. Fletcher R (1976) Conjugate gradient methods for indefinite systems, volume 506 of Lecture Notes Math. 73–89. Springer-Verlag Berlin, New York.Google Scholar
  7. Freund RW (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14:470–482.CrossRefGoogle Scholar
  8. Freund RW, Golub GH, Nachtigal NM (1991) Iterative solution of linear systems. Technical Report 91.21 RIACS NASA Ames Research Center, Mottet Field.Google Scholar
  9. Freund RW, Nachtigal NM (1991) QMR: A quasi minimal residual method for non- Hermitian linear systems. Numer. Math. 60:315–339.CrossRefGoogle Scholar
  10. Gambolati G, Paniconi C, Putti M (1993) Mass transfer analysis in sorbing porous media by an integro-differential approach. In: Wang SSY (ed) Advances in Hydro-Science and -Engineering, Volume I, Part B The University of Mississippi, University, MS, pp 1819–1828.Google Scholar
  11. Gambolati G, Pini G, Putti M, Paniconi C (1994a) Finite element modeling of the transport of reactive contaminants in variably saturated soils with LEA and non-LEA sorption. In: Zannetti P (ed) Environmental Modelling II173–212. Computational Mechanics Publications Southampton, UK.Google Scholar
  12. Gambolati G, Gallo C, Paniconi C (1994b) Numerical integration methods for the dual porosity model in sorbing porous media. 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 621–628.Google Scholar
  13. Gambolati G, Gallo C, Paniconi C, Putti M (1995) Numerical solutions for nonequilibrium solute transport in porous media. In:Advances in Hydro-Science and -Engineering, Volume 1, Part B Tsinghua University Press, Beijing, China, pp 1733–1742.Google Scholar
  14. Gamerdinger AP, Wagenet RJ, van Genuchten MT (1990) Application of two-site/two- region models for studying simultaneous nonequilibrium transport and degradation of pesticides.Soil Sci. Soc. Amer. J. 54:957–963.CrossRefGoogle Scholar
  15. Hestenes MR, Stiefel E (1952) Methods of conjugate gradient for solving linear systems. J. Res. Nat. Bur. Standard 49:409–436.Google Scholar
  16. Kershaw DS (1978) The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations. J. Comp. Phys. 26:43–65.CrossRefGoogle Scholar
  17. Lanczos C (1952) Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bur. Standard 49:33–53.Google Scholar
  18. Leismann HM, Herrling B, Krenn V (1988) A quick algorithm for the dead-end pore concept for modeling large-scale propagation processes in groundwater. In: Celia MA, Ferrand L, Brebbia CA, Gray WG, Pinder GF (eds) Proceedings of the VII International Conference on Computational Methods in Water Resources, Vol. 2, Numerical Methods for Transport and Hydrologie Processes CMP Elsevier, Amsterdam, Holland, pp 275–280.Google Scholar
  19. Nkedi-Kizza P, Biggar JW, Selim HM, van Genuchten MT, Wierenga PJ, Davidson JM, Nielsen DR (1984) On the equivalence of two conceptual models for describing ion exchange during transport through an aggregated Oxisol. Water Resour. Res 20(8):1123–1130.CrossRefGoogle Scholar
  20. Pini G, Putti M (1994) Krylov methods in the finite element solution of groundwater transport problems. 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 1431–1438.Google Scholar
  21. Rao PSC, Davidson JM, Jessup RE, Selim HM (1979) Evaluation of conceptual models for describing nonequilibrium adsorption of pesticides during steady-flow in soils. Soil Sci. Soc. Amer. J. 42:22–28.CrossRefGoogle Scholar
  22. Reid JK (1971) On the method of conjugate gradients for the solution of large sparse systems of linear equations. In: Reid JK (ed)Large Sparse Sets of Linear Equations Academic Press, New York, pp 231–253.Google Scholar
  23. Saad Y (1981) Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp. 37(155):105–126.CrossRefGoogle Scholar
  24. Saad Y (1990) Krylov subspace methods: Theory, algorithms, and applications. In: Glowinski R, Lichnewsky A (eds) Computing Methods in Applied Sciences and Engineering SIAM, Philadelphia, pp 24–41.Google Scholar
  25. Saad Y (1991) ILUT: A dual strategy accurate incomplete ILU factorization. Technical report Minnesota Supercomputer Institute, University of Minnesota.Google Scholar
  26. Saad Y, Schultz MH (1985) Conjugate gradient-like algorithms for solving nonsymmetric linear systems. Math. Comp. 44(170):417–424.CrossRefGoogle Scholar
  27. Saad Y, Schultz MH (1986) GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3):856–869.CrossRefGoogle Scholar
  28. Selim HM, Davidson JM, Mansell RS (1976) Evaluation of a two-site adsorption- desorption model for describing solute transport in soils. In: Proc. of Summer Computer Simulation Conference Simulation Councils, La Jolla, CA, pp 444–448.Google Scholar
  29. Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10:36–52.CrossRefGoogle Scholar
  30. Sudicky EA (1990) The Laplace transform Galerkin technique for efficient time-continuous solution of solute transport in double-porosity media. Geoderma 46:209–232.CrossRefGoogle Scholar
  31. van der Vorst HA (1990) Iterative methods for the solution of large systems of equations on supercomputers. Adv. Water Resources 13(3): 137–146.CrossRefGoogle Scholar
  32. van der Vorst HA (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.CrossRefGoogle Scholar
  33. van Genuchten MT, Wierenga PJ (1976) Mass transfer studies in sorbing porous media: 1. Analytical solutions. Soil Sci. Soc. Amer. J. 40(4):473–480.CrossRefGoogle Scholar
  34. van Genuchten MT, Wagenet RJ (1989) Two-site/two-region models for pesticide transport and degradation: Theoretical development and analytical solutions. Soil Sci. Soc. Amer. J. 53:1303–1310.CrossRefGoogle Scholar
  35. Weber Jr. WJ, McGinley P, Katz L (1991) Sorption phenomena in subsurface system- s: Concepts, models and effects on contaminant fate and transport. Water Res. 25(5):499–528.CrossRefGoogle Scholar
  36. Young DM, Jea KC (1980) Generalized conjugate-gradient acceleration of nonsymmetriz- able iterative methods. Linear Algebra Appl. 34:159–194.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2010

Authors and Affiliations

  • Giuseppe Gambolati
    • 1
  • Mario Putti
    • 1
  • Claudio Paniconi
    • 2
  1. 1.Dipartimento di Metodi e Modelli Matematici per le Scienze ApplicateUniversity of PaduaPadovaItaly
  2. 2.CRS4CagliariItaly

Personalised recommendations