article

ε-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers

Authors:

Kaushik Mukherjee,

Srinivasan NatesanAuthors Info & Claims

Numerical Algorithms, Volume 58, Issue 1

Pages 103 - 141

https://doi.org/10.1007/s11075-011-9449-6

Published: 01 September 2011 Publication History

Abstract

This article is devoted to the study of a hybrid numerical scheme for a class of singularly perturbed parabolic convection-diffusion problems with discontinuous convection coefficients. In general, the solutions of this class of problems possess strong interior layers. To solve these problems, we discretize the time derivative by the backward-Euler method and the spatial derivatives by a hybrid finite difference scheme (a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions) on a layer resolving piecewise-uniform Shishkin mesh. It is proved that the method converges uniformly in the discrete supremum norm with almost second-order spatial accuracy. Moreover, an optimal order of convergence (up to a logarithmic factor) is obtained inside the layer regions. Extensive numerical experiments are conducted to support the theoretical results and also, to demonstrate the accuracy of this method.

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Cited By

Yadav NMukherjee K(2024)Efficient parameter-robust numerical methods for singularly perturbed semilinear parabolic PDEs of convection-diffusion typeNumerical Algorithms10.1007/s11075-023-01670-296:2(925-973)Online publication date: 1-Jun-2024
https://dl.acm.org/doi/10.1007/s11075-023-01670-2
Priyadarshana SMohapatra JRamos H(2023)Robust numerical schemes for time delayed singularly perturbed parabolic problems with discontinuous convection and source termsCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-023-00552-261:1Online publication date: 29-Nov-2023
https://dl.acm.org/doi/10.1007/s10092-023-00552-2
Das ANatesan S(2015)Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin meshApplied Mathematics and Computation10.1016/j.amc.2015.08.137271:C(168-186)Online publication date: 15-Nov-2015
https://dl.acm.org/doi/10.1016/j.amc.2015.08.137
Show More Cited By

ε-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis

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Published In

cover image Numerical Algorithms

Numerical Algorithms Volume 58, Issue 1

September 2011

138 pages

ISSN:1017-1398

Issue’s Table of Contents

Copyright © Copyright © 2011 Springer Science+Business Media, LLC.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 September 2011

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Cited By

Yadav NMukherjee K(2024)Efficient parameter-robust numerical methods for singularly perturbed semilinear parabolic PDEs of convection-diffusion typeNumerical Algorithms10.1007/s11075-023-01670-296:2(925-973)Online publication date: 1-Jun-2024
https://dl.acm.org/doi/10.1007/s11075-023-01670-2
Priyadarshana SMohapatra JRamos H(2023)Robust numerical schemes for time delayed singularly perturbed parabolic problems with discontinuous convection and source termsCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-023-00552-261:1Online publication date: 29-Nov-2023
https://dl.acm.org/doi/10.1007/s10092-023-00552-2
Das ANatesan S(2015)Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin meshApplied Mathematics and Computation10.1016/j.amc.2015.08.137271:C(168-186)Online publication date: 15-Nov-2015
https://dl.acm.org/doi/10.1016/j.amc.2015.08.137
Mukherjee KNatesan S(2012)An Efficient Hybrid Numerical Scheme for Singularly Perturbed Problems of Mixed Parabolic-Elliptic TypeRevised Selected Papers of the 5th International Conference on Numerical Analysis and Its Applications - Volume 823610.1007/978-3-642-41515-9_46(411-419)Online publication date: 15-Jun-2012
https://dl.acm.org/doi/10.1007/978-3-642-41515-9_46

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