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Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems

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Abstract

The aim of this paper is to investigate whether we can accelerate the order of convergence of existing high order methods to solve some singularly perturbed two-point BVPs. To this end, we consider a fitted mesh finite difference method of Patidar (Appl. Math. Comput., 188:720–733, 2007) applied on a mesh of Shishkin type for the solution of self-adjoint problem which is ε-uniformly convergent of order four. We attempted to increase the order of convergence by Richardson’s extrapolation and discovered that this well-known convergence acceleration technique has some limitations. We observe that even though this extrapolation technique improves the accuracy slightly, it does not increase the rate of convergence which is originally four for the underlying method for the problem above. Theoretical investigations are demonstrated by some numerical experiments.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, 7535, South Africa

    Justin B. Munyakazi & Kailash C. Patidar

Authors
  1. Justin B. Munyakazi
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  2. Kailash C. Patidar
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Correspondence to Kailash C. Patidar.

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The research contained in this paper was supported by the South African National Research Foundation.

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Munyakazi, J.B., Patidar, K.C. Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems. J. Appl. Math. Comput. 32, 219–236 (2010). https://doi.org/10.1007/s12190-009-0245-6

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  • DOI: https://doi.org/10.1007/s12190-009-0245-6

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