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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.

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