## Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem

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**48**(1987), 551-564 Request permission

## Abstract:

A family of uniformly accurate finite-difference schemes for the model problem $- \varepsilon u”+ a(x)u’+ b(x)u = f(x)$ is constructed using a general finite-difference framework of Lynch and Rice [*Math. Comp.*, v. 34, 1980, pp. 333-372] and Doedel [

*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 450-465], A scheme of order ${h^p}$ (uniform in $\varepsilon$) is constructed to be exact on a collection of functions of the type $\{ 1,x, \ldots ,{x^p},\exp (\frac {1}{\varepsilon }\smallint a),x\exp (\frac {1}{\varepsilon }\smallint a), \ldots ,{x^{p - 1}}\exp (\frac {1}{\varepsilon }\smallint a)\}$. The high order is achieved by using extra evaluations of the source term

*f*. The details of the construction of such a scheme (for general

*p*) and a complete discretization error analysis, which uses the stability results of Niederdrenk and Yserentant [

*Numer. Math.*, v. 41, 1983, pp. 223-253], are given. Numerical experiments exhibiting uniform orders ${h^p}$, $p = 1,2,3, \text {and}\;4$, are presented.

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## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp.
**48**(1987), 551-564 - MSC: Primary 65L10; Secondary 34B05, 34E15
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878690-0
- MathSciNet review: 878690