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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