61 research outputs found
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
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
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
Efficient implementation of geometric integrators for separable Hamiltonian problems
We here investigate the efficient implementation of the energy-conserving
methods named Hamiltonian Boundary Value Methods (HBVMs) recently introduced
for the numerical solution of Hamiltonian problems. In this note, we describe
an iterative procedure, based on a triangular splitting, for solving the
generated discrete problems, when the problem at hand is separable.Comment: 4 page
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