A minimal-variable symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic

On an asymptotic method for computing the modified energy for symplectic methods

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory

Long-time averaging for integrable Hamiltonian dynamics

International audienceGiven a Hamiltonian dynamics, we address the question of computing the space-average (referred as the ensemble average in the field of molecular simulation) of an observable through the limit of its time-average. For a completely integrable system, it is known that ergodicity can be characterized by a diophantine condition on its frequencies and that the two averages then coincide. In this paper, we show that we can improve the rate of convergence upon using a filter function in the time-averages. We then show that this convergence persists when a numerical symplectic scheme is applied to the system, up to the order of the integrator