Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

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

The Lack of Continuity and the Role of Infinite and Infinitesimal in Numerical Methods for ODEs: the Case of Symplecticity

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 page

Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods

In this paper we are concerned with the analysis of a class of geometric integrators, at first devised in [14, 18], which can be regarded as an energy-conserving variant of Gauss collocation methods. With these latter they share the property of conserving quadratic first integrals but, in addition, they also conserve the Hamiltonian function itself. We here reformulate the methods in a more convenient way, and propose a more refined analysis than that given in [18] also providing, as a by-product, a practical procedure for their implementation. A thorough comparison with the original Gauss methods is carried out by means of a few numerical tests solving Hamiltonian and Poisson problems.Comment: 28 pages, 2 figures, 4 table

Numerical comparisons among some methods for Hamiltonian problems

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 figure

On the Existence of Energy-Preserving Symplectic Integrators Based upon Gauss Collocation Formulae

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 modifie

The Hamiltonian BVMs (HBVMs) Homepage

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.htmlComment: 49 pages, 16 figures; Chapter 4 modified; minor corrections to Chapter 5; References update