In this paper, we provide a simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods. The framework consists in coupling two simple tools: firstly a local Fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge–Kutta methods can be derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see [2] and references therein). A few numerical tests are also included, in order to confirm the effectiveness of the methods resulting from our analysis.

A simple framework for the derivation and analysis of effective one-step methods for ODEs / L.Brugnano; F.Iavernaro; D.Trigiante. - In: APPLIED MATHEMATICS AND COMPUTATION. - ISSN 0096-3003. - STAMPA. - 218:(2012), pp. 8475-8485. [10.1016/j.amc.2012.01.074]

A simple framework for the derivation and analysis of effective one-step methods for ODEs

BRUGNANO, LUIGI;TRIGIANTE, DONATO
2012

Abstract

In this paper, we provide a simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods. The framework consists in coupling two simple tools: firstly a local Fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge–Kutta methods can be derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see [2] and references therein). A few numerical tests are also included, in order to confirm the effectiveness of the methods resulting from our analysis.
2012
218
8475
8485
L.Brugnano; F.Iavernaro; D.Trigiante
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/596211
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