Razafindralandy et al. Adv. Model. and Simul. in Eng. Sci. (2019)6:5 https://doi.org/10.1186/s40323-019-0130-2 R E V I E W Open Access Some robust integrators for large time dynamics Dina Razafindralandy1* , Vladimir Salnikov2, Aziz Hamdouni1 and Ahmad Deeb1 *Correspondence: dina.razafi
[email protected] Abstract 1Laboratoire des Sciences de l’Ingénieur pour l’Environnement This article reviews some integrators particularly suitable for the numerical resolution of (LaSIE), UMR 7356 CNRS, differential equations on a large time interval. Symplectic integrators are presented. Université de La Rochelle, Their stability on exponentially large time is shown through numerical examples. Next, Avenue Michel Crépeau, 17042 La Rochelle Cedex, France Dirac integrators for constrained systems are exposed. An application on chaotic Full list of author information is dynamics is presented. Lastly, for systems having no exploitable geometric structure, available at the end of the article the Borel–Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, Long-time stability, Borel summation, Divergent series Introduction In many domains of mechanics, simulations over a large time interval are crucial. This is, for instance, the case in molecular dynamics, in weather forecast or in astronomy. While many time integrators are available in literature, only few of them are suitable for large time simulations. Indeed, many numerical schemes fail to correctly predict the expected physical phenomena such as energy preservation, as the simulation time grows. For equations having an underlying geometric structure (Hamiltonian systems, varia- tional problems, Lie symmetry group, Dirac structure, ...), geometric integrators appear to be very robust for large time simulation.