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A note concerning the El-Mistikawy Werle exponential finite difference scheme for a boundary turning point problem. (English) Zbl 0673.65046
BAIL III, Proc. 3rd Int. Conf. Boundary and interior layers, Dublin/Ireland 1984, Conf. Ser. 6, 145-150 (1984).
Summary: [For the entire collection see Zbl 0671.00015.]
The author, H. Han and R. B. Kellogg [(*) Computational and asymptotic methods for boundary and interior layer, Boole Pres. Conf. Ser. 4, 13-27 (1982; Zbl 0511.65063) and (**) Math. Comput. 42, 465-492 (1984; Zbl 0542.34050)] considered a modified version of the exponential three point finite difference scheme by T. M. El-Mistikawy and M. J. Werle [AIAA J. 16, 749-751 (1978; Zbl 0383.76018)] for numerical solution of the problem $Ly \equiv -\varepsilon y_{xx}(x) - p(x)y_ x(x) + q(x)y(x)=f(x) \text{ for }0<x<1,$ $$y(0)=d_1$$ and $$y(1)=d_2$$, where $$p(z)=0$$ for some $$z$$ in $$[0,1)$$. Here $$\varepsilon$$ is a constant in $$(0,1]$$, the functions $$p$$, $$q$$, and $$f$$ are “sufficiently smooth”, $$q(x)>0$$ on $$[0,1]$$, the constants $$d_1$$ and $$d_2$$ are given boundary data, and, for the case $$z=0$$, $$d_1$$ is assumed to be within $$O(\varepsilon^{\beta /2})$$ of $$f(0)/q(0)$$ where $$\beta \equiv q(z)/p_ x(z)>0$$. Error estimates obtained in (*) and (**) for the El-Mistikawy Werle scheme indicated that a certain modification of this schemed neard thed turning point $$z$$ would prevent degradation of accuracy near $$z$$ when a certain function $$\phi$$ is small. Numerical results in (**) suggested that for $$z$$  in $$(0,1)$$ the unmodified El-Mistikawy Werle scheme indeed suffers this loss of accuracy, while numerical experiments in (*) for the case $$z=0$$ displayed no such deterioration in accuracy. In this note analysis will be provided which substantiates the observation that when $$z=0$$ the unmodified El-Mistikawy Werle scheme suffers no such loss of accuracy in comparison with the modified version.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations