The European Physical Journal Special Topics

, Volume 226, Issue 15, pp 3303–3314 | Cite as

A numerical study of nonlinear diffusion phenomena in heterogeneous media: energy transfer at diverse blow-up modes and self-organisation processes

  • Yuri N. Skiba
  • Denis M. FilatovEmail author
Regular Article
Part of the following topical collections:
  1. Challenges in the Analysis of Complex Systems: Prediction, Causality and Communication


A detailed analysis of a new method for numerical simulation of nonlinear diffusion phenomena is carried out. The method is based on operator splitting performed in time and space, and yields highly accurate solutions in complex 2D and 3D computational domains. After providing a circumstantial mathematical description of the developed method, we test it in several numerical experiments aimed, firstly, to model energy transfer at diverse modes of evolution of the dynamical system, and, secondly, to simulate self-organisation processes typical for real-world applications. A discussion of the outcomes of the numerical experiments is given. This is a follow-up paper of our recent original results presented at the 19th European conference on mathematics for industry.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Bear, Dynamics of fluids in porous media (Dover Publications, New York, 1988) Google Scholar
  2. 2.
    D.A. Frank-Kamenetsky, Diffusion of heat transfer in chemical kinetics (Nauka, Moscow, 1967) (in Russian) Google Scholar
  3. 3.
    M.E. Glicksman, Diffusion in solids: field theory, solid-state principles and applications (John Wiley & Sons, New York, 2000) Google Scholar
  4. 4.
    S.P. Kurdyumov, Regimes with blow-up (Fizmatlit, Moscow, 2006) (in Russian) Google Scholar
  5. 5.
    P.V. Makarov, Phys. Mesomech. 13, 292 (2010) CrossRefGoogle Scholar
  6. 6.
    H. Mehrer, Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes (Springer-Verlag, Berlin, 2007) Google Scholar
  7. 7.
    A.Kh. Vorob’yov, Diffusion problems in chemical kinetics (Moscow University Press, Moscow, 2003) (in Russian) Google Scholar
  8. 8.
    Z. Wu, J. Zhao, J. Yin, H. Li, Nonlinear diffusion equations (World Scientific Publishing, Singapore, 2001) Google Scholar
  9. 9.
    A.A. Samarskii et al., Blow-up in quasilinear parabolic equations (Walter de Gruyter, Berlin, 1995) Google Scholar
  10. 10.
    F. Gibou, R. Fedkiw, J. Comput. Phys. 202, 577 (2005) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Necati Özişik, Finite difference methods in heat transfer (CRC Press, Boca Raton, 1994) Google Scholar
  12. 12.
    A.A. Samarskii et al., in Blow-up modes: an evolution of the idea. Co-evolution laws of complex structures (Nauka, Moscow, 1999), pp. 39–46 (in Russian) Google Scholar
  13. 13.
    G.P. Bystrai, I.A. Lykov, S.A. Okhotnikov, 2011, arXiv:1109.5019 []
  14. 14.
    A.V. Gurevich, A.N. Karashtin, Phys. Rev. Lett. 110, 185005 (2013) ADSCrossRefGoogle Scholar
  15. 15.
    I.A. Lykov, Ph.D. thesis, Ekaterinburg, Russia, 2013 (in Russian) Google Scholar
  16. 16.
    Yu.N. Skiba, D.M. Filatov, in Advances in mathematics research (Nova Science Publishers, USA, 2013), Vol. 18, pp. 271–298 Google Scholar
  17. 17.
    Yu.N. Skiba, D.M. Filatov, Appl. Math. Comput. 219, 8467 (2013) MathSciNetGoogle Scholar
  18. 18.
    T.D. Lee, Mathematical methods in physics (Columbia University, New York, 1964) Google Scholar
  19. 19.
    G.I. Marchuk, Methods of computational mathematics (Springer-Verlag, New York, 1982) Google Scholar
  20. 20.
    S.V. Tsynkov, Appl. Numer. Math. 27, 465 (1998) MathSciNetCrossRefGoogle Scholar
  21. 21.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes: the art of scientific computing (Cambridge University Press, Cambridge, 2007) Google Scholar
  22. 22.
    P.D. Lax, R.D. Richtmyer, Commun. Pure Appl. Math. 9, 267 (1956) CrossRefGoogle Scholar
  23. 23.
    J.C. Strikwerda, Finite difference schemes and partial differential equations (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2004) Google Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Centre for Atmospheric Sciences (CCA), National Autonomous University of Mexico (UNAM)Mexico CityMexico
  2. 2.Sceptica Scientific LtdStockportUK

Personalised recommendations