2,759 research outputs found
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 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
Fifty Years of Stiffness
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 figure
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
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
Conservative precision agriculture: first economic and energetic assessments within the Agricare project
The integration of conservation tillage techniques with the principles of site-specific management characterising precision agriculture is an innovative feature aimed to achieve better economic and environmental sustainability, increasingly required by Community agricultural policies
A Two Step, Fourth Order, Nearly-Linear Method with Energy Preserving Properties
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 table
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