Order conditions for General Linear Nyström methods

• Published in: Numerical algorithms Vol. 65; no. 3; pp. 579 - 595
• Main Authors: D’Ambrosio, Raffaele, De Martino, Giuseppe, Paternoster, Beatrice
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2014; Springer Nature B.V
• Subjects: Algebra; Algorithms; Boundary value problems; Computer Science; Differential equations; Economic conditions; Initial value problems; Numeric Computing; Numerical Analysis; Numerical methods; Original Paper; Representations; Theory of Computation; 65L05; Order conditions; Second order ordinary differential equations; B-series; General linear methods; Nyström methods
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-013-9819-3

The purpose of this paper is to analyze the algebraic theory of order for the family of general linear Nyström (GLN) methods introduced in D’Ambrosio et al. (Numer. Algorithm 61 (2), 331–349, 2012 ) with the aim to provide a general framework for the representation and analysis of numerical methods solving initial value problems based on second order ordinary differential equations (ODEs). Our investigation is carried out by suitably extending the theory of B-series for second order ODEs to the case of GLN methods, which leads to a general set of order conditions. This allows to recover the order conditions of numerical methods already known in the literature, but also to assess a general approach to study the order conditions of new methods, simply regarding them as GLN methods: the obtained results are indeed applied to both known and new methods for second order ODEs.