43 research outputs found
A note on the efficient implementation of Hamiltonian BVMs
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 figure
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
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
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
Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles
Gyrocenter dynamics of charged particles plays a fundamental role in plasma
physics. In particular, accuracy and conservation of energy are important
features for correctly performing long-time simulations. For this purpose, we
here propose arbitrarily high-order energy conserving methods for its
simulation. The analysis and the efficient implementation of the methods are
fully described, and some numerical tests are reported.Comment: 23 pages, 4 figure
Efficient implementation of Radau collocation methods
In this paper we define an efficient implementation of Runge-Kutta methods of
Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems.
The proposed implementation relies on an alternative low-rank formulation of
the methods, for which a splitting procedure is easily defined. The linear
convergence analysis of this splitting procedure exhibits excellent properties,
which are confirmed by its performance on a few numerical tests.Comment: 19 pages, 3 figures, 9 table
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
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
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