Skip to main content
Springer Nature Link
Log in

An adaptive numerical method to handle blow-up in a parabolic system

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Abia, L.M., López-Marcos, J.C., Martínez, J.: On the blow-up time convergence of semi-discretizations of reaction-diffusion equations. Appl. Numer. Math. 26(4), 399–414 (1998)

    Article  Google Scholar 

  2. Acosta, G., Fernández Bonder, J., Groisman, P., Rossi, J.D.: Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions. M2AN Math. Model. Numer. Anal. 36(1), 55–68 (2002)

  3. Bandle, C., Brunner, H.: Blowup in diffusion equations: a survey. J. Comput. Appl. Math. 97(1–2), 3–22 (1998)

    Google Scholar 

  4. Berger, M., Kohn, R.V.: A rescaling algorithm for the numerical calculation of blowing-up solutions. Comm. Pure Appl. Math. 41(6), 841–863 (1988)

    Google Scholar 

  5. Brändle, C., Groisman, P., Rossi, J.D.: Fully discrete adaptive methods for a blow-up problem. Math. Models Methods Appl. Sci. 14(10), 1425–1450 (2004)

    Article  Google Scholar 

  6. Brändle, C., Quirós, F., Rossi, J.D.: Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal. To appear

  7. Budd, C.J., Huang, W., Russell, R.D.: Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17(2), 305–327 (1996)

    Article  Google Scholar 

  8. Chlebík, M., Fila, M.: Some recent results on blow-up on the boundary for the heat equation. In: Evolution equations: existence, regularity and singularities (Warsaw, 1998), 61–71. Banach Center Publ. vol. 52, Polish Acad. Sci. Warsaw. 2000

  9. Duran, R.G., Etcheverry, J.I., Rossi, J.D.: Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete Contin. Dynam. Systems 4(3), 497–506 (1998)

    Google Scholar 

  10. Fernández Bonder, J., Rossi, J.D.: Blow-up vs. spurious steady solutions. Proc. Amer. Math. Soc. 129(1), 139–144 (2001)

    Google Scholar 

  11. Ferreira, R., Groisman, P., Rossi, J.D.: Numerical blow-up for a nonlinear problem with a nonlinear boundary condition. Math. Models Methods Appl. Sci. 12(4), 461–483 (2002)

    Article  Google Scholar 

  12. Ferreira, R., Groisman, P., Rossi, J.D.: Adaptive numerical schemes for a parabolic problem with blow-up. IMA J. Numer. Anal. 23(3), 439–463 (2003)

    Article  Google Scholar 

  13. Fila, M., Filo, J.: Blow-up on the boundary: a survey. In: Singularities and differential equations (Warsaw, 1993) 67–78. Banach Center Publ. vol. 33, Polish Acad. Sci. Warsaw. 1996

  14. Filo, J.: Diffusivity versus absorption through the boundary. J. Differential Equations 99(2), 281–305 (1992)

    Article  Google Scholar 

  15. Groisman, P., Quirós, F., Rossi, J.D.: Non-simultaneous blow-up in a numerical approximation of a parabolic system. Comput. Appl. Math. 21(3), 813–831 (2002)

    Google Scholar 

  16. Nakagawa, T.: Blowing up of a finite difference solution to u t =u xx +u2. Appl. Math. Optim. 2(4), 337–350 (1975/76)

    Article  Google Scholar 

  17. Pedersen, M., Lin, Z.: Blow-up estimates of the positive solution of a parabolic system. J. Math. Anal. Appl. 255(2), 551–563 (2001)

    Article  Google Scholar 

  18. Quirós, F., Rossi, J.D.: Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys. 52(2), 342–346 (2001)

    Google Scholar 

  19. Quirós, F., Rossi, J.D.: Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud. 3(3), 397–418 (2003)

    Google Scholar 

  20. Rossi, J.D.: The blow-up rate for a system of heat equations with non-trivial coupling at the boundary. Math. Methods Appl. Sci. 20 (1), 1–11 (1997)

    Article  Google Scholar 

  21. Souplet, P., Tayachi, S.: Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J. Math. Soc. Japan 56(2), 571–584 (2004)

    Google Scholar 

  22. Wang, M.: Fast-slow diffusion systems with nonlinear boundary conditions. Nonlinear Anal. Ser. A: Theory Methods, 46(6), 893–908 (2001)

    Google Scholar 

  23. Wang, M., Wang, S.: Quasilinear reaction-diffusion systems with nonlinear boundary conditions. J. Math. Anal. Appl. 231(1), 21–33 (1999)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, U. Autónoma de Madrid, 28049, Madrid, Spain

    Cristina Brändle & Fernando Quirós

  2. IMAFF, CSIC, c/ Serrano 123, 28006, Madrid, Spain

    Julio D. Rossi

  3. Departamento de Matemática, FCEyN, U. de Buenos Aires, Ciudad Universitaria Pab. I, (1428), Buenos Aires, Argentina

    Julio D. Rossi

Authors
  1. Cristina Brändle
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Fernando Quirós
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Julio D. Rossi
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Cristina Brändle.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brändle, C., Quirós, F. & D. Rossi, J. An adaptive numerical method to handle blow-up in a parabolic system. Numer. Math. 102, 39–59 (2005). https://doi.org/10.1007/s00211-005-0638-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0638-x

Mathematics Subject Classification (2000)