93 research outputs found
Enhanced HBVMs for the numerical solution of Hamiltonian problems with multiple invariants
Recently, the class of energy-conserving Runge-Kutta methods named
Hamiltonian Boundary Value Methods (HBVMs), has been proposed for the efficient
solution of Hamiltonian problems, as well as for other types of conservative
problems. In this paper, we report further advances concerning such methods,
resulting in their enhanced version (Enhanced HBVMs, or EHBVMs). The basic
theoretical results are sketched, along with a few numerical tests on a
Hamiltonian problem, taken from the literature, possessing multiple invariants.Comment: 4 page
Blended General Linear Methods based on Boundary Value Methods in the GBDF family
Among the methods for solving ODE-IVPs, the class of General Linear Methods
(GLMs) is able to encompass most of them, ranging from Linear Multistep
Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able
to overcome typical drawbacks of the previous classes of methods. For example,
order barriers for stable LMF and the problem of order reduction for RK
methods. Nevertheless, these goals are usually achieved at the price of a
higher computational cost. Consequently, many efforts have been made in order
to derive GLMs with particular features, to be exploited for their efficient
implementation. In recent years, the derivation of GLMs from particular
Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has
been proposed for the numerical solution of stiff ODE-IVPs. In particular, this
approach has been recently developed, resulting in a new family of L-stable
GLMs of arbitrarily high order, whose theory is here completed and fully
worked-out. Moreover, for each one of such methods, it is possible to define a
corresponding Blended GLM which is equivalent to it from the point of view of
the stability and order properties. These blended methods, in turn, allow the
definition of efficient nonlinear splittings for solving the generated discrete
problems. A few numerical tests, confirming the excellent potential of such
blended methods, are also reported.Comment: 22 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
Parallel Factorizations in Numerical Analysis
In this paper we review the parallel solution of sparse linear systems,
usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is
based on the concept of parallel factorization of a (block) tridiagonal matrix.
This allows to obtain efficient parallel extensions of many known matrix
factorizations, and to derive, as a by-product, a unifying approach to the
parallel solution of ODEs.Comment: 15 pages, 5 figure
Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
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 versio
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
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