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.
Similar content being viewed by others
References
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Linß, T., Madden, N.: Accurate solution of a system of coupled singularly perturbed reaction-diffusion equations. Computing 73, 121–133 (2004)
Linß, T., Madden, N.: Layer-adapted meshes for a system of coupled singularly perturbed reaction-diffusion problems. IMA J. Numer. Anal. (2008, in press)
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)
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)
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)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs (1967)
Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and Flow Problems. Springer, New York (1996)
Shishkin, G.I.: Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural section, Ekaterinburg (1992) (in Russian)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Martin Stynes.
This research was partially supported by a grant from EUROPA XXI of the Caja de Ahorros de la Inmaculada and the project of the University of Zaragoza UZ2006-CIE-09.
Rights and permissions
About this article
Cite this article
Gracia, J.L., Lisbona, F.J. & O’Riordan, E. A coupled system of singularly perturbed parabolic reaction-diffusion equations. Adv Comput Math 32, 43 (2010). https://doi.org/10.1007/s10444-008-9086-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-008-9086-3
Keywords
- Singularly perturbed parabolic reaction-diffusion equations
- Layer-adapted piecewise-uniform mesh
- Singular perturbation parameters