The application of implicit Runge-Kutta and collection methods to boundary-value problems

Author:
Richard Weiss

Journal:
Math. Comp. **28** (1974), 449-464

MSC:
Primary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1974-0341881-2

MathSciNet review:
0341881

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Abstract | References | Similar Articles | Additional Information

Abstract: The solution of a nonlinear system of first order differential equations with nonlinear boundary conditions by implicit Runge-Kutta methods based on interpolatory quadrature formulae is examined. An equivalence between implicit Runge-Kutta and collocation schemes is established. It is shown that the difference equations obtained have a unique solution in a neighbourhood of an isolated solution of the continuous problem, that this solution can be computed by Newton iteration and that it converges to the isolated solution. The order of convergence is equal to the degree of precision of the related quadrature formula plus one. The efficient implementation of the methods is discussed and numerical examples are given.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0341881-2

Keywords:
Implicit Runge-Kutta method,
collocation method,
boundary-value problem

Article copyright:
© Copyright 1974
American Mathematical Society