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Solution of nonlinear convection-diffusion problems by a conservative Galerkin-characteristics method

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Abstract

We consider a scheme for nonlinear (degenerate) convection dominant diffusion problems that arise in contaminant transport in porous media with equilibrium adsorption isotherm. This scheme is based on a regularization relaxation scheme that has been introduced by Jäger and Kačur (Numer Math 60:407–427, 1991; M2AN Math Model Numer Anal 29(N5):605–627, 1995) with a type of numerical integration by Bermejo (SIAM J Numer Anal 32:425–455, 1995) to the modified method of characteristics with adjusted advection MMOCAA that was recently developed by Douglas et al. (Numer Math 83(3):353–369, 1999; Comput Geosci 1:155–190, 1997). We present another variant of adjusting advection method. The convergence of the scheme is proved. An error estimate of the approximated scheme is derived. Computational experiments are carried out to illustrate the capability of the scheme to conserve the mass.

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References

  1. Arbogast, T., Huang, C.-s.: A full mass and volume conserving implementation of a characteristic method for transport problems. SIAM J. Sci. Comput. 28, 2001 (2006). ICES Report 05-08 (2005)

    Google Scholar 

  2. Arbogast T., Wheeler M.F.: A characteristics-mixed finite element method for advection dominated transport problems. SIAM J. Numer. Anal. 32, 404–424 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barrett J., Knabner P.: An improved error bound for a Lagrange–Galerkin method for contaminant transport with non-lipschitzian adsorption kinetics. SIAM J. Numer. Anal. 5, 1862–1883 (1998)

    Article  MathSciNet  Google Scholar 

  4. Bear J.: Dynamics of Fluid in Porous Media, 2nd edn. Elsevier, New York (1972)

    Google Scholar 

  5. Bermejo R.: A Galerkin-characteristics algorithm for transport-diffusion equation. SIAM J. Numer. Anal. 32, 425–455 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Blazy, S., Marquardt, O.: A characteristic algorithm for the 3D Navier–Stokes equation using padfem. In: Proceedings of the Conference Parallel and Distributed Computing and Systems (2003)

  7. Cannon, J.R.: The One-dimensional Heat Equation. Encyclopedia of Mathematics and its Applications, vol. 23. Addison-Wesley, Reading (1984)

  8. Celia M.A., Russell T.F., Herrera I., Ewing R.E.: An Eulerian–Lagrangian localized adjoint method for the advection-diffusion equation. Adv. Water Resour. 13(N4), 187–206 (1990)

    Article  Google Scholar 

  9. Chen Z., Ewing R., Jiang Q., Spagnuolo A.: Error analysis for Characteristics-based methods for degenerate parabolic problems. SIAM J. Numer. Anal. 40, 1491–1515 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Dawson C.N., van Duijn C.J., Grundy R.E.: Large time asymptotic in contaminant transport in porous media. SIAM J. Appl. Math. 56(N4), 965–993 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dawson C.N.: Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equation. SIAM J. Numer. Anal. 5, 1709–1725 (1998)

    Article  Google Scholar 

  13. Douglas J., Russell T.F.: Numerical methods for convection dominated diffusion problems based on combining the method of the characteristics with finite elements or finite differences. SIAM J. Numer. Anal. 19, 871–885 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Douglas J., Huang C.-s., Pereira F.: The modified method of characteristics with adjusted advection. Numer. Math. 83(3), 353–369 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Douglas J., Furtado F., Pereira F.: On the numerical simulation of water flooding of heterogeneous petroleum reseviors. Comput. Geosci. 1, 155–190 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Feistauer M.: Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 67. Longman Scientific & Technical Harlow, UK (1993)

    Google Scholar 

  17. Giles C.H., Smith D., Huitson A.: A general treatment and classification of the solute adsorption isotherm, I. Theoretical. J. Colloidal Surf. Sci. 47, 755–765 (1974)

    Google Scholar 

  18. Grundy R.E., van Duijn C.J., Dawson C.N.: Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media: the fast reaction case. Q. J. Mech. Appl. Math. 12, 69–106 (1994)

    Article  Google Scholar 

  19. Jäger W., Kačur J.: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. M2AN Math. Model. Numer. Anal. 29(N5), 605–627 (1995)

    MATH  Google Scholar 

  20. Kačur, J.: Method of Rothe in Evolution Equations, vol. 80. Teubner-Texte zur Mathematic, Leipzig (1985)

  21. Kačur J.: Solution to strongly nonlinear parabolic problems by a linear approximation scheme. IMA J. Numer. Anal. 19, 1–26 (1996)

    Google Scholar 

  22. Kačur J.: Solution of degenerate convection-diffusion problems by the method of characteristics. SIAM J. Numer. Anal. 39, 858–879 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kačur J., Mahmood M.Sh.: Solution of Solute transport in unsaturated porous media by the by the method of characteristics. Numer. Methods Partial Differ. Equ. 19, 732–761 (2003)

    Article  MATH  Google Scholar 

  24. Kačur J., Mahmood M.Sh.: Galerkin characteristics method for convection-diffusion problems with memory terms. Int. J. Numer. Anal. Model. 6(1), 89–109 (2009)

    MATH  Google Scholar 

  25. Karlsen K.H., Lie K.-A.: Unconditionally stable splitting scheme for a class of nonlinear parabolic equations. IMA J. Numer. Anal. 19(4), 609–635 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Knabner, P.: Mathematische Modelle für transport und sorption gelöster stoffe in porösen medien. Methoden und Verfahren der Mathematischen Physik, 36 (1991)

  27. Knabner, P.: Finite-element-approximation of solute transport in porous media with general adsorption processes. In: Xiao, S. (ed.) Flow and Transport in Porous Media, Summer school, Beijing, 8–26 August 1988, pp. 223–292. World Scientific, Singapore (1992)

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

  29. Kufner A., John O., Fučí k S.: Function Spaces. Nordhoff, Leiden (1977)

    MATH  Google Scholar 

  30. Mahmood, M.Sh.: The regularized Galerkin characteristics algorithm for contaminant transport with equilibrium and non-equilibrium adsorption. Ph.D. thesis, Comenius University (2002)

  31. Mahmood, M.Sh.: Analysis of Galerkin-characteristics algorithm for variably saturated flow in porous media, CMMSE (2006)

  32. Minev P.D., Ethier C.R.: A characteristic/finite element algorithm for the 3D Navier–Stokes equations using unstructured grids. Comput. Method Appl. M 178, 39–50 (1999)

    Article  MATH  Google Scholar 

  33. Morton K.W., Priestley A., Süli: Stability of the Lagrange–Galerkin method with non-exact integration. RAIRO model. Math. Anal. Numer. 4, 225–250 (1988)

    Google Scholar 

  34. Nec̆as J.: Les méthodes directes en théorie des élliptiques. Academia, Prague (1967)

    Google Scholar 

  35. Pironneau O.: On the transport-diffusion algorithm and its applications to the Naiver–Stokes equations. Numer. Math. 38, 309–332 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  36. Reible D.: Fundamentals of Environmental Engineering. Springer, Berlin (1999)

    Google Scholar 

  37. Simon J.: Compact sets in the space L p (0, T ; B). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  38. van Duijn C.J., Knabner P.: Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption: traveling waves. J. Reine Angewandte Math. 415, 1–49 (1991)

    MATH  Google Scholar 

  39. van Duijn C.N., Knabner P.: Travelling waves in the transport of reactive solutes through porous media: adsorption and ion exchange—Part 1. Transp. Porous Media 8, 167–226 (1992)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Mechanical Engineering Faculty, University of Žilina, 010 26, Zilina, Slovakia

    Mohammed Shuker Mahmood

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  1. Mohammed Shuker Mahmood
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Correspondence to Mohammed Shuker Mahmood.

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This work was supported in part by the scientific grant agency of the Ministry of Education of the Slovak Republic (ME SR) and of Slovak Academy of Sciences (SAS). 1/0843/08 (21).

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Mahmood, M.S. Solution of nonlinear convection-diffusion problems by a conservative Galerkin-characteristics method. Numer. Math. 112, 601–636 (2009). https://doi.org/10.1007/s00211-009-0221-y

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  • DOI: https://doi.org/10.1007/s00211-009-0221-y

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