FM 13-14 (arXiv:1410.5982)

Authors: Michele Bartuccelli, Jonathan Deane, Guido Gentile

Title: The high-order Euler method and the spin-orbit model
 
Abstract: We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin-orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an excellent measure of agreement between analytical predictions and the proposed fast algorithm.

Keywords: Fast numerics for differential equation; spin-orbit model; periodic attractors; quasi-periodic attractors; series expansion; perturbation theory.

Michele V. Bartuccelli
Department of Mathematics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: gentile@mat.uniroma3.it

Jonathan H.B. Deane
Department of Mathematics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: j.deane@surrey.ac.uk

Guido Gentile
Dipartimento di Matematica e Fisica
Università Roma Tre
Largo San Leonardo Murialdo 1 - 00146 Roma - Italy
e-mail: gentile@mat.uniroma3.it