article

HODIE finite difference schemes on generalized Shishkin meshes

Authors:

C. Clavero,

J. L. GraciaAuthors Info & Claims

Journal of Computational and Applied Mathematics, Volume 164-165, Issue 1

Pages 195 - 206

https://doi.org/10.1016/S0377-0427(03)00653-8

Published: 01 March 2004 Publication History

Abstract

In this work we study a class of HODIE finite difference schemes to solve linear one-dimensional convection-diffusion problems of singular perturbation type. The numerical method is constructed on nonuniform Shishkin type meshes, defined by a generating function, including classical Shishkin meshes and Shishkin-Bakhvalov meshes. We will prove the uniform convergence, with respect to the singular perturbation parameter, of the HODIE scheme on this type of meshes, having order bigger than one. We show some numerical examples confirming in practice the theoretical results and also we see numerically that an appropriate extrapolation will be useful to improve the errors and the order of convergence, when the singular perturbation parameter is sufficiently small.

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Cited By

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Das P(2019)Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problemsJournal of Computational and Applied Mathematics10.1016/j.cam.2015.04.034290:C(16-25)Online publication date: 3-Jan-2019
https://dl.acm.org/doi/10.1016/j.cam.2015.04.034
Kumar SKumar M(2014)High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delayComputers & Mathematics with Applications10.1016/j.camwa.2014.09.00468:10(1355-1367)Online publication date: 1-Nov-2014
https://dl.acm.org/doi/10.1016/j.camwa.2014.09.004
Mukherjee KNatesan S(2011)Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficientsBIT10.1007/s10543-010-0292-251:2(289-315)Online publication date: 1-Jun-2011
https://dl.acm.org/doi/10.1007/s10543-010-0292-2

Index Terms

HODIE finite difference schemes on generalized Shishkin meshes
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations

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cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 164-165, Issue 1

Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)

1 March 2004

799 pages

ISSN:0377-0427

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Elsevier Science Publishers B. V.

Netherlands

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Published: 01 March 2004

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Das P(2019)Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problemsJournal of Computational and Applied Mathematics10.1016/j.cam.2015.04.034290:C(16-25)Online publication date: 3-Jan-2019
https://dl.acm.org/doi/10.1016/j.cam.2015.04.034
Kumar SKumar M(2014)High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delayComputers & Mathematics with Applications10.1016/j.camwa.2014.09.00468:10(1355-1367)Online publication date: 1-Nov-2014
https://dl.acm.org/doi/10.1016/j.camwa.2014.09.004
Mukherjee KNatesan S(2011)Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficientsBIT10.1007/s10543-010-0292-251:2(289-315)Online publication date: 1-Jun-2011
https://dl.acm.org/doi/10.1007/s10543-010-0292-2

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Abstract

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Cited By

Index Terms

Recommendations

High order methods for elliptic and time dependent reaction-diffusion singularly perturbed problems

Uniformly convergent additive finite difference schemes for singularly perturbed parabolic reaction-diffusion systems

On the hybrid finite difference scheme for a singularly perturbed Riccati equation

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