Skip to main content
SpringerLink
Log in

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the Courant-Friedrichs-Lewy (CFL) condition failed).

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

Similar content being viewed by others

References

  1. Babušikova, J.: Application of relaxation scheme to degenerate variational inequalities. Appl. Math. 46, 419–439 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bänsch, E.: Numerical experiments with adaptivity for the porous medium equation. Acta Math. Univ. Comen. (N.S.) 64, 425–454 (1995)

    Google Scholar 

  3. Barrett, J., Kappmeier, H., Knabner P.: Lagrange-Galerkin approximation for advection-dominated nonlinear contaminant transport in porous media. In: Computational Methods in Water Resources X, vol. 1, pp. 299–307. (1994)

  4. Barrett, J., Kappmeier, H., Knabner, P.: Lagrange-Galerkin approximation for advection-dominated contaminant transport with nonlinear equilibrium or non-equilibrium adsorption. In: Helmig, R., et al. (ed.) Modelling and Computations in Environmental Science, pp. 36–48. Vieweg, Braunschweig (1997)

    Google Scholar 

  5. Dafermos, C.: Polygonal approximation of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dawson, C.: Analysis of an upwind – mixed finite element method for nonlinear contaminant transport equations. SIAM J. Numer. Anal. 35(5), 1709–1729 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dawson, C., Van Duijn, C., Grundy, R.: Large time asymptotics in contaminant transport in porous media. SIAM J. Appl. Math. 56, 965–993 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dawson, C., Van Duijn, C., Wheeler, M.: Characteristic-Galerkin methods for contaminant transport with non-equilibrium adsorption kinetics. SIAM J. Numer. Anal. 31(4), 982–999 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frolkovič, P., Geiser, J.: Numerical simulations of radionuclides transport in double porosity media with sorption. In: Handlovicova, A., et al. (ed.) Algoritmy 2000, pp. 28–36. Slovak University of Technology, Bratislava (2000)

    Google Scholar 

  10. Grundy, R., Van Duijn, C., Dawson, C.: Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media. Q. J. Mech. Appl. Math. 1(47), 69–106 (1994)

    Article  Google Scholar 

  11. Handlovičova, A.: Solution of Stefan problems by fully discrete linear schemes. Acta Math. Univ. Comen., N.S. 67(2), 351–372 (1998)

    MATH  Google Scholar 

  12. Holden, H., Karlsen, K., Lie, K.-A.: Operator splitting methods for degenerate convection–diffusion equations: II. Numerical examples with emphasis on reservoir simulation and sedimentation. Comput. Geosci. 4, 287–323 (2000)

    Article  MATH  Google Scholar 

  13. Jäger, W., Kačur, J.: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Model. Numer. Anal. 29, 605–627 (1995)

    MATH  Google Scholar 

  14. Kačur, J.: Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Num. Anal. 19, 119–154 (1999)

    Article  MATH  Google Scholar 

  15. Kačur, J., van Keer, R.: Solution of contaminant transport with adsorption in porous media by the method of characteristics. M2AN 35(5), 981–1006 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kačur, J., Frolkovič, P.: Semi-analytical solutions for contaminant transport with nonlinear sorption in 1D. Preprint 24, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Heidelberg (2002)

  17. Kružkov, S.: First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10(2), 217–243 (1970)

    Article  Google Scholar 

  18. LeVeque, R.J.: Numerical methods for conservation laws, Lectures in Mathematics. ETH Zürich, Birkhäuser-Verlag, Basel (1992)

  19. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002)

  20. Maple 9.5, http://www.maplesoft.com

  21. Mikula, K.: Numerical solution of nonlinear diffusion with finite extinction phenomena. Acta Math. Univ. Comen., N.S. 2, 223–292 (1995)

    MathSciNet  Google Scholar 

  22. Ohlberger, M.: Higher order finite volume methods on self-adaptive grids for convection dominated reactive transport problems in porous media. Comput. Vis. Sci. 7(1), 41–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Olejnik, O.: Discontinuous solutions of non-linear differential equations. Am. Math. Soc. Transl. 2(26), 95–172 (1963)

    Google Scholar 

  24. Olejnik, O.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Am. Math. Soc. Transl. 2(33), 285–290 (1963)

    Google Scholar 

  25. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press (1992)

  26. Sheng, D., Smith, D.W.: Analytical solutions to the advective contaminant transport equation with nonlinear sorption. Int. J. Numer. Anal. Methods Geomech. 23, 853–879 (1999)

    Article  MATH  Google Scholar 

  27. Van Duijn, C., Knabner, P.: Solute transport in porous media with equilibrium and nonequilibrium multiple site adsorption: Traveling waves. J. Reine Angew. Math. 415, 1–49 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Simulation in Technology Center, University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    Peter Frolkovič

  2. Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynska dolina, 842 48, Bratislava, Slovakia

    Jozef Kačur

Authors
  1. Peter Frolkovič
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jozef Kačur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Frolkovič.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frolkovič, P., Kačur, J. Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D. Comput Geosci 10, 279–290 (2006). https://doi.org/10.1007/s10596-006-9023-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-006-9023-9

Keywords

AMS Subject Classification

Search

Navigation