Abstract
In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed fromthe backward Euler scheme to amonotone hybrid scheme. Three kinds of nonuniform grids are considered, especially the standard Shishkin mesh, the Bakhavalov–Shishkinmesh and the adaptive grid. Themethods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and second order for themonotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal.
Similar content being viewed by others
References
Amiraliyev, G.M. and Duru, H., A Note on a Parameterized Singular Perturbation Problem, J. Comp. Appl. Math., 2005, vol. 182, pp. 233–242.
Amiraliyev, G.M., Kudu, M., and, Duru H., Uniform Difference Method for a Parameterized Singular Perturbation Problem, Appl. Math. Comp., 2006, vol. 175, pp. 89–100.
Beckett, G. and Mackenzie, J.A., Convergence Analysis of Finite Difference Approximations on Equidistributed Grids to a Singularly Perturbed Boundary Value Problem, Appl. Num. Math., 2000, vol. 35, no. 106, pp. 87–109.
de Boor C., Good Approximation by Splines with Variable Knots, Spline Functions and Approximation Theory, Proc. Symp. held at the University of Alberta, Edmonton, May 29-June 1, 1972, Meir, A. and Sharma, A. (Eds.) Basel: Birkhauser, 1973.
Cen, Z., A Second-Order Difference Scheme for a Parameterized Singular Perturbation Problem, J. Comp. Appl. Math., 2008, vol. 221, pp. 174–182.
Das, P. and Natesan, S., Numerical Solution of a System of Singularly Perturbed Convection Diffusion Boundary Value Problems Using Mesh Equidistribution Technique, Aust. J. Math. An. Appl., 2013, vol. 10, pp. 1–17.
Das, P. and Natesan, S., Optimal Error Estimate Using Mesh Equidistribution Technique for Singularly Perturbed System of Reaction-Diffusion Boundary-Value Problems, Appl. Math. Comp., 2014, vol. 249, pp. 265–277.
Chen, Y., Uniform Convergence Analysis of Finite Difference Approximations for Singular Perturbation Problems on an Adapted Grid, Adv. Comp. Math., 2006, vol. 24, pp. 197–212.
Farrell, P.A., Hegarty, A.F, Miller, J.M., O’Riordan, E., and Shishkin, G.I., Robust Computational Techniques for Boundary Layers, Boca Raton, FL: Chapman and Hall/CRC Press, 2000.
Feckan, M., Parametrized Singularly Perturbed Boundary Value Problems, J. Math. An. Appl., 1994, vol. 188, pp. 426–435.
Jankowski, T. and Lakshmikantham, V, Monotone Iterations for Differential Equations with a Parameter, J. Appl. Math. Stoch. An., 1997, vol. 10, no. 3, pp. 273–278.
Kellogg, R.B. and Tsan, A., Analysis of Some Difference Approximations fora Singular Perturbation Problem without Turning Points, Math. Comp., 1978, vol. 32, pp. 1025–1039.
Kopteva, N. and Stynes, M., A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem, SIAM J. Num. An., 2001, vol. 39, pp. 1446–1467.
Kopteva, N., Madden, N., and Stynes, M., Grid Equidistribution for Reaction-Diffusion Problems in One Dimension, Num. Alg., 2005, vol. 40, pp. 305–322.
Liu, X. and McRae, FA., A Monotone Iterative Method for Boundary Value Problems of Parametric Differential Equations, J. Appl. Math. Stoch. An., 2001, vol. 14, no. 2, pp. 183–187.
Linß, T., Sufficient Conditions for Uniform Convergence on Layer-Adapted Grids, Appl. Num. Math., 2001, vol. 37, pp. 241–255.
Liseikin, VD., Grid Generation Methods, Berlin: Springer, 1999.
Liseikin, V.D., Layer Resolving Grids and Transformations for Singular Perturbation Problems, VSP, 2001.
Mackenzie, J., Uniform Convergence Analysis of an Upwind Finite-Difference Approximation of a Convection-Diffusion Boundary Value Problem on an Adaptive dgrid, IMA J. Num. An., 1999, vol. 19, pp. 233–249.
Miller, J.J.H., O’Riordan, E., and Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems, Singapore: World Scientific, 2012.
Mohapatra, J. and Natesan, S., Parameter-Uniform Numerical Method for Global Solution and Global Normalized Flux of Singularly Perturbed Boundary Value Problems Using Grid Equidistribution, Comp. Math. Appl., 2010, vol. 60, pp. 1924–1939.
Pomentale, T., A Constructive Theorem of Existence and Uniqueness for Problem y' = f(x, y, λ), y(a) = α, y(b) = β, Z. Angrew. Math. Mech., 1976, vol. 56, no. 8, pp. 387/388.
Ronto, M. and Csikos-Marinets, T., On the Investigation of Some Non-Linear Boundary Value Problems with Parameters, Math. Notes, Miskolc, 2000, vol. 1, no. 2, pp. 157–166.
Roos, H.G., Stynes, M., and Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations, Berlin: Springer, 2008.
Shakti, D. and Mohapatra, J., A Second Order Numerical Method for a Class of Parameterized Singular Perturbation Problems on Adaptive Grid, Nonlin. Eng., 2017, vol. 6, no. 3, pp. 221–228.
Turkyilmazoglu, M., Analytic Approximate Solutions of Parameterized Unperturbed and Singularly Perturbed Boundary Value Problems, Appl. Math. Model., 2011, vol. 35, pp. 3879–3886.
Xie, F., Wang, J., Zhang, W., and He, M., A Novel Method for a Class of Parameterized Singularly Perturbed Boundary Value Problems, J. Comp. Appl. Math., 2008, vol. 213, pp. 258–267.
Xu, X., Huang, W., Russell, R.D., and Williams, J.F, Convergence of de Boor’s Algorithm for the Generation of Equidistributing Meshes, IMA J. Num. An., 2011, vol. 31, pp. 580–596.
Acknowledgements
The authors express their sincere thanks to the anonymous reviewers for their valuable comments and suggestions to improve the quality and presentation of the manuscript.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 2, pp. 213–228.
Rights and permissions
About this article
Cite this article
Shakti, D., Mohapatra, J. Parameter-Uniform Numerical Methods for a Class of Parameterized Singular Perturbation Problems. Numer. Analys. Appl. 12, 176–190 (2019). https://doi.org/10.1134/S1995423919020071
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995423919020071