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

Keywords: Ordinary differential equations Runge–Kutta methods One-step methods Hamiltonian problems Hamiltonian Boundary Value Methods Energy preserving methods Symplectic methods Energy drift a b s t r a c t 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. One-step methods are widely used in the numerical solution of initial value problems for ordinary differential equations which, without loss of generality, we shall assume to be in the form: y 0 ðtÞ ¼ f ðyðtÞÞ; t 2 ½0; TŠ; yð0Þ ¼ y 0 2 R m : ð1Þ In particular, we consider a very general class of effective one-step methods that can be led back to a local Fourier expansion of the continuous problem over the interval [0,h], where h is the considered stepsize. In general, different choices of the basis result in different classes of methods, for which, however, the analysis turns out to be remarkably simple. Though the arguments can be extended to a general choice of the basis, we consider here only the case of a polynomial basis, obtaining a large subclass of Runge–Kutta methods, even though trigonometric or exponential fitted type bases (see, e.g. [1,9,14,16,18]) could be, in principle, considered. Usually, the order properties of such methods are studied through the classical theory of Butcher on rooted trees (see, e.g. [8, Chapter 3]), almost always resorting to the so called simplifying assumptions (see, e.g. [8, Section 321]). For the methods derived in the new framework (see Section 2), such analysis turns out to …