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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Collocation methods for boundary value problems on “long” intervals
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by Peter A. Markowich and Christian A. Ringhofer PDF
Math. Comp. 40 (1983), 123-150 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Math. Comp. 40 (1983), 123-150
  • MSC: Primary 65L10; Secondary 65D07
  • DOI: https://doi.org/10.1090/S0025-5718-1983-0679437-X
  • MathSciNet review: 679437