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High order generalized upwind schemes and numerical solution of singular perturbation problems. (English) Zbl 1121.65089

This paper is concerned with the numerical solution of second order singularly perturbed two–point boundary value problems that can be written in the form: \[ \varepsilon y''(x) = f(x, y(x), y'(x)), x \in [a,b],\;0 < \varepsilon <<1, \] with \( y(a)=y_a, y(b)=y_b\) where \(f\) is a sufficiently smooth function.
The main difference with the standard approach is that here instead of transforming the second order differential system into an equivalent first order system, the authors propose to discretize directly the second order equation by using \(k\)-step finite difference schemes with order \(k\) given by the same authors [J. Comput. Appl. Math. 176, 59–76 (2005; Zbl 1073.65061)] together with upwind techniques to avoid the undesirable oscillations in the boundary layer. Some examples are presented to illustrate the effect of the upwind in removing the spurious oscillations.
Finally, the numerical results with three test problems are given to show the behavior of different strategies of step size variation together with discretizations of different orders on the global error of these problems.

MSC:

65L12 Finite difference and finite volume methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations

Citations:

Zbl 1073.65061

Software:

MIRKDC; bvp4c; PMIRKDC
PDFBibTeX XMLCite
Full Text: DOI

References:

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