An Hermite–Obreshkov method for 2nd-order linear initial-value problems for ODE

The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if o...

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Veröffentlicht in:Numerical algorithms Jg. 96; H. 3; S. 1109 - 1141
1. Verfasser: Corless, Robert M
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer Nature B.V 01.07.2024
Schlagworte:
Algorithms
Differential equations
Linear equations
Mathematicians
Methods
Ordinary differential equations
Polynomials
Taylor series
ISSN:1017-1398, 1572-9265
Online-Zugang:Volltext
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Zusammenfassung:The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if only a single scalar equation such as the Mathieu equation y′′+(a-2qcos2x)y=0 is being considered. Nonetheless, the topic is not yet exhausted, and this paper considers the case of writing an efficient arbitrary-precision code for the solution of such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearly spectral accuracy at a cost only polynomial in the number of bits of accuracy requested. This is interesting for the Mathieu equation in particular because the solutions can be highly oscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method and summarizes the approximation theoretic results justifying the choice of a balanced Hermite–Obreshkov method including its backward stability and decent Lebesgue constants. This method may be of especial interest for the solution of so-called D-finite equations, for which Taylor series coefficients up to degree m are available at cost only O(m), instead of the more usual O(m2). This paper celebrates the happy occasion of the 90th birthday of John C. Butcher.
Bibliographie:ObjectType-Article-1
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01738-z