Abstract
We examine a time dependent singularly perturbed convection-diffusion problem, where the convective coefficient contains an interior layer. A smooth transformation is introduced to align the grid to the location of the interior layer. A numerical method consisting of an upwinded finite difference operator and a piecewise-uniform Shishkin mesh is constructed in this transformed domain. Numerical results are presented which indicate that the numerical approximations converge at a rate of first order (up to logarithmic factors) uniformly in the pointwise maximum norm.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Dunne, R.K., O’ Riordan, E.: Interior layers arising in linear singularly perturbed differential equations with discontinuous coefficients. In: Farago, I., Vabishchevich P., Vulkov, L. (eds.) Proceedings of Fourth International Conference on Finite Difference Methods: Theory and Applications, pp. 29–38. Rousse University, Bulgaria (2007)
Dunne, R.K., O’ Riordan, E., Shishkin, G.I.: A fitted mesh method for a class of singularly perturbed parabolic problems with a boundary turning point. Comput. Methods Appl. Math. 3(3), 361–372 (2003)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’ Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. CRC, Boca Raton (2000)
Gracia, J.L., O’ Riordan, E.: A singularly perturbed convection–diffusion problem with a moving interior layer. Int. J. Numer. Anal. Model. 9(4), 823–843 (2012)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Transactions of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)
O’ Riordan, E., Quinn, J.: A linearized singularly perturbed convection–diffusion problem with an interior layer. Appl. Numer. Math. 98, 1–17 (2015)
O’ Riordan, E., Shishkin, G.I.: Singularly perturbed parabolic problems with non-smooth data. J. Comput. Appl. Math. 166(1), 233–245 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
O’Riordan, E., Quinn, J. (2015). Numerical Experiments with a Linear Convection–Diffusion Problem Containing a Time-Varying Interior Layer. In: Knobloch, P. (eds) Boundary and Interior Layers, Computational and Asymptotic Methods - BAIL 2014. Lecture Notes in Computational Science and Engineering, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-319-25727-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-25727-3_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25725-9
Online ISBN: 978-3-319-25727-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)