Hamiltonian BVMs (HBVMs): A Family of “Drift Free” Methods for Integrating polynomial Hamiltonian problems

Abstract

In this paper, we present a number of numerical results concerning the newly introduced class of Hamiltonian Boundary Value Methods (hereafter, HBVMs). Such methods are very suited for the numerical integration of Hamiltonian problems, since they are able to preserve, in the discrete solution, the exact value of polynomial Hamiltonians. In such a way, a numerical drift of the Hamiltonian, sometimes experienced when solving such problems, cannot occur.

A simple framework for the derivation and analysis of effective one-step methods for ODEs

“…Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge-Kutta methods can be derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see [1] and references therein). A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods.…”

Recent Advances on Preserving Methods for Poisson Systems

“…At the best of our knowledge, the first instance of energy-preserving Runge-Kutta methods for polynomial Hamiltonian systems was given by Iavernaro and Pace [10] and, as observed in [7], they can also be derived by the discretization of the averaged vector field method in [13]. HBVMs were formerly presented in [4] (see also [2] and [3] and references therein), as a generalization of the s-Stage Trapezoidal Methods [10] and of the energy-preserving methods for polynomial Hamiltonians derived in [11,12]. For Poisson type systems, in a recent paper of Cohen and Hairer [8], which, in turn, generalize the formulae presented in [9], some energy-preserving methods are proposed.…”

“…They are also able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see [3] and references therein). The proposed methods turn out to be equivalent to those recently derived in [8], giving therefore an alternative point of view that provides additional insight on the methods.…”

Recent Advances on Preserving Methods for General Conservative Systems

“…We consider the following methods of order four: GHBVM(10,2) described above, which yields a conservation of the three first integrals within machine precision; the energy-conserving method HBVM (10,2), and the (symplectic) Gauss method of order four (GAUSS4). For comparison reasons we also consider the Symplectic Euler method and the Störmer-Verlet method.…”

“…More recently, energy preserving Runge-Kutta methods of order two have been derived in [12], based on the concept of discrete line integral. This idea, further developed, has led to fourth order examples of conservative Runge-Kutta methods [13,14] and, finally, to Hamiltonian Boundary Value Methods (HBVMs) [3,2,4,5,6,7,8], a class of energy-preserving Runge-Kutta methods of any high order. Even though energy-conservation is an important feature for the discrete dynamical system induced by the methods, many Hamiltonian problems (and, in general, conservative problems) are characterized by the presence of multiple invariants.…”