Advertisement

A coupled system of singularly perturbed parabolic reaction-diffusion equations

  • J. L. Gracia
  • F. J. Lisbona
  • E. O’Riordan
Article

Abstract

In this paper systems with an arbitrary number of singularly perturbed parabolic reaction-diffusion equations are examined. A numerical method is constructed for these systems which involves an appropriate layer-adapted piecewise-uniform mesh. The numerical approximations generated from this method are shown to be uniformly convergent with respect to the singular perturbation parameters. Numerical experiments supporting the theoretical results are given.

Keywords

Singularly perturbed parabolic reaction-diffusion equations Layer-adapted piecewise-uniform mesh Singular perturbation parameters 

Mathematics Subject Classifications (2000)

65M15 35K50 35K57 

References

  1. 1.
    Bakhvalov, N.S.: On the optimization of methods for boundary-value problems with boundary layers. J. Numer. Methods Math. Phys. 9, 841–859 (1969) (in Russian)zbMATHGoogle Scholar
  2. 2.
    Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24, 1286–1303 (1960)zbMATHCrossRefGoogle Scholar
  3. 3.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman and Hall/CRC Press, Boca Raton (2000)zbMATHGoogle Scholar
  4. 4.
    Gracia, J.L., Lisbona, F.J.: A uniformly convergent scheme for a system of reaction-diffusion equations. J. Comput. Appl. Math. 206, 1–16 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gracia, J.L., Lisbona, F.J., O’Riordan, E.: A system of singularly perturbed reaction-diffusion equations. Dublin City University preprint MS–07–10 (2007)Google Scholar
  6. 6.
    Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: ε–uniform schemes with high order time–accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal. 20, 99–121 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R.B. Kellogg, Madden, N., Stynes, M.: A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions. Numer. Methods Partial Differential Equations 24, 312–334 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kellogg, R.B., Linß, T., Stynes, M.: A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions. Math. Comp. (2008, in press)Google Scholar
  9. 9.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasilinear equations of parabolic type. In: Translations of Mathematical Monographs, vol 23. American Mathematical Society, Providence (1968)Google Scholar
  10. 10.
    Linß, T., Madden, N.: An improved error estimate for a numerical method for a system of coupled singularly perturbed reaction-diffusion equations. Comput. Methods Appl. Math. 3, 417–423 (2003)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Linß, T., Madden, N.: A finite element analysis of a coupled system of singularly perturbed reaction-diffusion equations. Appl. Math. Comput. 148, 869–880 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Linß, T., Madden, N.: Accurate solution of a system of coupled singularly perturbed reaction-diffusion equations. Computing 73, 121–133 (2004)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Linß, T., Madden, N.: Layer-adapted meshes for a system of coupled singularly perturbed reaction-diffusion problems. IMA J. Numer. Anal. (2008, in press)Google Scholar
  14. 14.
    Madden, N., Stynes, M.: A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. IMA J. Numer. Anal. 23, 627–644 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Matthews, S., O’ Riordan, E., Shishkin, G.I.: A numerical method for a system of singularly perturbed reaction-diffusion problems. J. Comput. Appl. Math. 145, 151–166 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted mesh methods for the singularly perturbed reaction-diffusion problem. In: Minchev, E. (ed.) V-th International Conference on Numerical Analysis, August, 1996, pp. 99–105. Academic Publications, Plovdiv (1997)Google Scholar
  17. 17.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs (1967)Google Scholar
  18. 18.
    Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and Flow Problems. Springer, New York (1996)zbMATHGoogle Scholar
  19. 19.
    Shishkin, G.I.: Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural section, Ekaterinburg (1992) (in Russian)Google Scholar
  20. 20.
    Shishkin, G.I.: Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations. Comput. Math. Math. Phys. 35, 429–446 (1995)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Shishkin, G.I.: Approximation of a system of singularly perturbed elliptic reaction–diffusion equations on a rectangle. In: Farago, I., Vabishchevich, P., Vulkov, L. (eds.) Fourth International Conference on Finite Difference Methods: Theory and Applications, August, 2006, pp. 125–133. Rousse University, Lozenetz (2007)Google Scholar
  22. 22.
    Shishkina, L., Shishkin, G.I.: Robust numerical method for a system of singularly perturbed parabolic reaction–diffusion equations on a rectangle. Math. Model. Anal. 13, 251–261 (2008)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Departamento de Matemática AplicadaUniversidad de ZaragozaZaragozaSpain
  2. 2.School of Mathematical SciencesDublin City UniversityDublin 9Ireland

Personalised recommendations