Abstract. In this paper, we provide a simple framework to derive and analyse several classes of effective onestep methods. The framework consists in the discretization of a local Fourier expansion of the continuous problem. 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 [1] and references therein). A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods.

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We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems, via their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.Comment: 10 pages, 4 figure

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. For example, it is well known that classical symplectic methods can only exactly preserve, at most, quadratic Hamiltonians. In this paper, a new family of methods, called Hamiltonian Boundary Value Methods (HBVMs), is introduced and analyzed. HBVMs are able to exactly preserve, in the discrete solution, Hamiltonian functions of polynomial type of arbitrarily high degree. These methods turn out to be symmetric, precisely A-stable, and can have arbitrarily high order. A few numerical tests confirm the theoretical results.

In this paper, we present a number of numerical results concerning the newly introduced class of Hamiltonian Boundary Value Methods (hereafter, HBVMs). Such methods are very suited for the numerical integration of Hamiltonian problems, since they are able to preserve, in the discrete solution, the exact value of polynomial Hamiltonians. In such a way, a numerical drift of the Hamiltonian, sometimes experienced when solving such problems, cannot occur.