Numerical implementation of the asymptotic theory for classical diffusion in heterogeneous media

Abstract

Based on the asymptotic theory of impurity transport developed by one of the authors (P.S.K.), numerical calculations of the concentration for classical diffusion in heterogeneous media in one and two dimensions are performed. In parallel, for the same media, a direct numerical solution of the diffusion equation was carried out. The results of the two calculations are highly consistent with each other at asymptotically far distances from the impurity source. The computation time according to the asymptotic theory turned out to be two orders of magnitude less than the time required for direct calculations.

Graphic abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]

References

  1. 1.

    M.B. Isichenko, Rev. Mod. Phys. 64, 961 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    J.P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, 2000), p. 315

  4. 4.

    D.S. Dean, I.T. Drummond, R.R. Horgan, Journal of Statistical Mechanics: Theory and Experiment 2007, P07013 (July 2007)

  5. 5.

    D.M. Tartakovsky, M. Dentz, Transport in Porous Media 130, 105–127 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    L.A. Bolshov, P.S. Kondratenko, L.V. Matveev, Physics Uspekhi 62, 649–659 (2019)

    ADS  Article  Google Scholar 

  7. 7.

    P. Kekäläinen, M. Voutilainen, A. Poteri, P. Hölttä, A. Hautojärvi, J. Timonen, Transport Porous Med 87, 125–149 (2011)

    Article  Google Scholar 

  8. 8.

    Ch. Dong, S. Sun, G.A. Taylor, J. Porous Media 14(3), 219–242 (2011)

    Article  Google Scholar 

  9. 9.

    C. Masciopinto, G. Passarella, J. Contam. Hydrol. 215, 21–28 (2018)

    ADS  Article  Google Scholar 

  10. 10.

    Peter S. Kondratenko, Leonid V. Matveev, Alexander D. Vasiliev, J. Numer. Anal. Math. Modelling 34(6), 339–351 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    P.S. Kondratenko, JETP Letters 106(9), 604–607 (2017)

    ADS  Article  Google Scholar 

  12. 12.

    L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media (Oxford Butterworth Heinemann, 2004), Translated from Russian: Elektrodinamika Sploshnykh Sred (Moscow, Fizmatlit, 2005)

  13. 13.

    L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Oxford Butterworth Heinemann, 2004), Translated from Russian: Kvantovaya Mekhanika: Nerelyativiskaya Teoriya (Moscow, Fizmatlit, 2005)

  14. 14.

    P.S. Kondratenko, A.L. Matveev, JETP 130(4), 591–593 (2020)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

The authors are deeply grateful to Prof. L.V. Matveev for a fruitful discussion of the results. This work was financially supported by the Russian Science Foundation Grant no. 18-19-00533.

Author information

Affiliations

Authors

Contributions

PSK and ALM performed analytical calculations, ADV realized numerical computations. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to Peter S. Kondratenko.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kondratenko, P.S., Matveev, A.L. & Vasiliev, A.D. Numerical implementation of the asymptotic theory for classical diffusion in heterogeneous media. Eur. Phys. J. B 94, 50 (2021). https://doi.org/10.1140/epjb/s10051-020-00021-7

Download citation