"Fourier expansions and line integral methods for conservative problems"
Luigi Brugnano
Recently, the class of Hamiltonian Boundary Value Methods (HB-
VMs) [1] has been introduced with the aim of preserving the energy
associated with polynomial Hamiltonian systems (and, more in gen-
eral, with all suitably regular Hamiltonian systems).
HBVMs form a special subclass of Runge-Kutta methods which is easily introduced and
analysed by the aid of a local Fourier expansion of the continuous problem. This will be the subject of the present talk.
Many interesting problems admit other invariants besides the Hamiltonian
function. It would be therefore useful to have methods able to pre-
serve any number of independent invariants. This goal is here achieved
by generalizing the line-integral approach which HBVMs rely on, thus
obtaining a number of generalizations which we collectively name Line
Integral Methods. In fact, it turns out that this approach is quite gen-
eral, so that it can be applied to any numerical method whose discrete
solution can be suitably associated with a polynomial, such as a collo-
cation method, as well as to any conservative problem.
[1] The Hamiltonian BVMs (HBVMs) Homepage:
http://web.math.unifi.it/users/brugnano/HBVM/