Analisys of Hamiltonian Boundary Value Methods (HBVMs) for the numerical solution of polynomial Hamiltonian dynamical systems. (arXiv:0909.5659v1 [math.NA])

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 standard (even
symplectic) methods can only exactly preserve 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, perfectly $A$-stable, and can
have arbitrarily high order. A few numerical tests confirm the theoretical
results.

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Original source : http://arxiv.org/abs/0909.5659...