Runge-Kutta software with defect control for boundary value ODEs

A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggest...

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Bibliographic Details
Published in:SIAM journal on scientific computing 1996-03, Vol.17 (2), p.479-497
Main Authors: ENRIGHT, W. H, MUIR, P. H
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Codes
Computer science
Exact sciences and technology
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Numerical methods in mathematical programming
Numerical methods in mathematical programming, optimization and calculus of variations
Numerical methods in optimization and calculus of variations
Numerical methods in probability and statistics
Ordinary differential equations
Sciences and techniques of general use
Software
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Summary:A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge-Kutta formulas, known as mono-implicit Runge-Kutta (MIRK) formulas, which can be implemented at a lower cost per step than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather, only a discrete approximation at certain points within the problem interval is obtained. However, recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge-Kutta formula. These ideas have recently been extended to develop continuous extensions of the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on the continuous MIRK formulas, as an alternative to the standard use of global error control, as the basis for termination and mesh redistribution criteria.
ISSN:1064-8275
1095-7197
DOI:10.1137/S1064827593251496