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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Collocation methods for boundary value problems on “long” intervals

Authors: Peter A. Markowich and Christian A. Ringhofer
Journal: Math. Comp. 40 (1983), 123-150
MSC: Primary 65L10; Secondary 65D07
MathSciNet review: 679437
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Abstract: This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an A-stable symmetric collocation method. Problems arise, because standard theory predicts the use of many grid points as the length of the interval increases. Using the exponential decay of the ’infinite’ solution, an ’asymptotic’ a priori mesh-size sequence which increases exponentially, and which therefore only employs a reasonable number of meshpoints, is developed and stability, as the length of the interval tends to infinity, is shown. We also show that the condition number of the collocation equations is asymptotically proportional to the number of meshpoints employed when using this exponentially graded mesh. Using k-stage collocation at Gaussian points and requiring an accuracy $O(\varepsilon )$ at the knots implies that the number of meshpoints is $O({\varepsilon ^{ - 1/2k}})$ as $\varepsilon \to 0$.

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Keywords: Nonlinear boundary value problems, singular points, asymptotic properties, difference equations, stability of difference equations
Article copyright: © Copyright 1983 American Mathematical Society