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We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction term is O(h^5) (h is the stepsize of integration). The key tools the new methods are based upon are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results., Comment: 14 pages, 4 figures, 2 tables

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 figures

When numerically integrating canonical Hamiltonian systems, the long-term conservation of some of its invariants, among which the Hamiltonian function itself, assumes a central role. The classical approach to this problem has led to the definition of symplectic methods, among which we mention Gauss-Legendre collocation formulae. Indeed, in the continuous setting, energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations. However, this infinite process cannot be directly transferred to the discrete setting. By following a different approach, in this paper we describe a sequence of methods, sharing the same essential spectrum (and, then, the same essential properties), which are energy preserving starting from a certain element of the sequence on, i.e., after a finite number of steps., Comment: 15 pages

In this paper, we provide a simple framework to derive and analyse several classes of effective one-step 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 is 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. A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods., Comment: 11 pages, 2 figures (proofs of Thms. 2.2 and 3.1 simplified)

Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems. In the mainstream of this research, we have defined a new family of symplectic integrators depending on a real parameter. When it is zero, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null value of the parameter, the corresponding method remains symplectic and has order 2s-2: hence it may be interpreted as a symplectic perturbation of the Gauss method. Under suitable assumptions, it can be shown that the parameter a may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order 2s as the generating Gauss formula, and is able to preserve both energy and quadratic invariants., Comment: 4 pages

We report a few sumerical tests comparing some newly defined energy-preserving integrators and symplectic methods, using either constant and variable stepsize., Comment: 5 pages, 8 figures

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed., Comment: 25 pages, 4 figures (typos fixed)

We introduce a new family of symplectic integrators depending on a real parameter. When the paramer is zero, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null value of the parameter, the corresponding method remains symplectic and has order 2s-2: hence it may be interpreted as an order 2s-2 (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the free parameter may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting symplectic, energy conserving method shares the same order 2s as the generating Gauss formula., Comment: 19 pages, 7 figures; Sections 1, 2, and 6 sliglthly modified

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. Recently, a new class of methods, named Hamiltonian Boundary Value Methods (HBVMs) has been introduced and analysed, which are able to exactly preserve polynomial Hamiltonians of arbitrarily high degree. We here study a further property of such methods, namely that of having, when cast as a Runge-Kutta method, a matrix of the Butcher tableau with the same spectrum (apart from the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. This, in turn, allows to elucidate the existing connections with classical Runge-Kutta collocation methods., Comment: 12 pages

Hamiltonian Boundary Value Methods (in short, HBVMs) is a new class of numerical methods for the efficient numerical solution of canonical Hamiltonian systems. In particular, their main feature is that of exactly preserving, for the numerical solution, the value of the Hamiltonian function, when the latter is a polynomial of arbitrarily high degree. Clearly, this fact implies a practical conservation of any analytical Hamiltonian function. In this notes, we collect the introductory material on HBVMs contained in the HBVMs Homepage, available at http://web.math.unifi.it/users/brugnano/HBVM/index.html, Comment: 49 pages, 16 figures; Chapter 4 modified; minor corrections to Chapter 5; References updated

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. Recently, a new class of methods, named "Hamiltonian Boundary Value Methods (HBVMs)" has been introduced and analysed, which are able to exactly preserve polynomial Hamiltonians of arbitrarily high degree. We here study a further property of such methods, namely that of having, when cast as Runge-Kutta methods, a matrix of the Butcher tableau with the same spectrum (apart the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. Moreover, this allows their efficient "blended" implementation, for solving the generated discrete problems., Comment: 17 pages, 2 figures

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones., Comment: 24 pages, 11 figures

Recently, a new family of integrators (Hamiltonian Boundary ValueMethods) has been introduced, which is able to precisely conserve the energy function of polynomial Hamiltonian systems and to provide a practical conservation of the energy in the non-polynomial case. We settle the definition and the theory of such methods in a more general framework. Our aim is on the one hand to give account of their good behavior when applied to general Hamiltonian systems and, on the other hand, to find out what are the optimal formulae, in relation to the choice of the polynomial basis and of the distribution of the nodes. Such analysis is based upon the notion of extended collocation conditions and the definition of discrete line integral, and is carried out by looking at the limit of such family of methods as the number of the so called silent stages tends to infinity., Comment: 28 pages, 11 figures. Final revised version accepted for publication

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., Comment: 25 pages, 8 figures, revised version