Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the Tau method.

*(English)*Zbl 0574.65091This paper concerns the application of the second author’s recursive formulation of the Tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations. A practical error estimation technique, related to the concept of correction in the sense of P. Zadunaisky [Numer. Math. 27, 21-39 (1976; Zbl 0324.65035)], is considered and used in the design of an adaptive approach to the Tau method. It proves efficient in the numerical treatment of problems with rapid functional variations, stiff and singularly perturbed problems. A technique of increased accuracy at matching points of segmented Tau approximants is also discussed and successfully applied to several problems. Numerical examples show that, for a given degree of approximation, our segmented Tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials.

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34E15 | Singular perturbations, general theory for ordinary differential equations |