Computational Geosciences

, Volume 10, Issue 3, pp 279–290 | Cite as

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

  • Peter Frolkovič
  • Jozef Kačur


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).


contaminant transport finite volume method Freundlich isotherm nonlinear sorption 

AMS Subject Classification

35K65 35L67 65M25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babušikova, J.: Application of relaxation scheme to degenerate variational inequalities. Appl. Math. 46, 419–439 (2001)CrossRefMathSciNetMATHGoogle Scholar
  2. 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. 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)Google Scholar
  4. 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. 5.
    Dafermos, C.: Polygonal approximation of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)CrossRefMathSciNetMATHGoogle Scholar
  6. 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)CrossRefMathSciNetMATHGoogle Scholar
  7. 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)CrossRefMathSciNetMATHGoogle Scholar
  8. 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)CrossRefMathSciNetMATHGoogle Scholar
  9. 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. 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)CrossRefGoogle Scholar
  11. 11.
    Handlovičova, A.: Solution of Stefan problems by fully discrete linear schemes. Acta Math. Univ. Comen., N.S. 67(2), 351–372 (1998)MATHGoogle Scholar
  12. 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)CrossRefMATHGoogle Scholar
  13. 13.
    Jäger, W., Kačur, J.: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. Math. Model. Numer. Anal. 29, 605–627 (1995)MATHGoogle Scholar
  14. 14.
    Kačur, J.: Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Num. Anal. 19, 119–154 (1999)CrossRefMATHGoogle Scholar
  15. 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)CrossRefMathSciNetMATHGoogle Scholar
  16. 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)Google Scholar
  17. 17.
    Kružkov, S.: First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10(2), 217–243 (1970)CrossRefGoogle Scholar
  18. 18.
    LeVeque, R.J.: Numerical methods for conservation laws, Lectures in Mathematics. ETH Zürich, Birkhäuser-Verlag, Basel (1992)Google Scholar
  19. 19.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002)Google Scholar
  20. 20.
  21. 21.
    Mikula, K.: Numerical solution of nonlinear diffusion with finite extinction phenomena. Acta Math. Univ. Comen., N.S. 2, 223–292 (1995)MathSciNetGoogle Scholar
  22. 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)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Olejnik, O.: Discontinuous solutions of non-linear differential equations. Am. Math. Soc. Transl. 2(26), 95–172 (1963)Google Scholar
  24. 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. 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)Google Scholar
  26. 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)CrossRefMATHGoogle Scholar
  27. 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)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Simulation in Technology CenterUniversity of HeidelbergHeidelbergGermany
  2. 2.Department of Mathematical Analysis and Numerical MathematicsComenius UniversityBratislavaSlovakia

Personalised recommendations