Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance
Abstract
Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. This article proposes for arbitrary Hamiltonians similar integrators, which are explicit, of any even order, symplectic in an extended phase space, and with pleasant long time properties. They are based on a mechanical restraint that binds two copies of phase space together. Using backward error analysis, KolmogorovArnoldMoser theory, and additional multiscale analysis, an error bound of O (T δ^{l}ω ) is established for integrable systems, where T ,δ ,l , and ω are, respectively, the (long) simulation time, step size, integrator order, and some binding constant. For nonintegrable systems with positive Lyapunov exponents, such an error bound is generally impossible, but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear Schrödinger equation.
 Publication:

Physical Review E
 Pub Date:
 October 2016
 DOI:
 10.1103/PhysRevE.94.043303
 arXiv:
 arXiv:1609.02212
 Bibcode:
 2016PhRvE..94d3303T
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Dynamical Systems;
 Physics  Classical Physics;
 Physics  Computational Physics
 EPrint:
 Accepted by Phys. Rev. E