Symplectic integrators with adaptive time steps

Abstract

Abstract. In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, ∆ = ∆(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of ∆(t) and the orbital motion. In the second category the time step is a function of phase space variables ∆ = ∆(q, p). In this case, the system of equations to be solved is analyzed by introducing a new time variable τ with dt = ∆(q, p)dτ. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant ∆. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.

Analysis of the Chaotic Behavior of the Lower Hybrid Wave Propagation in Magnetised Plasma by Hamiltonian Theory

“…To solve this equation, let us callṼ = V − V and expand S 1 andṼ in a Fourier series over the angles: (34) so that the solution of Equation (33) becomes: (35) and the new variables are given by:…”

“…In particular, Figure 1 of [34] is analogous to our Figure 10. More recently, several different algorithms have been developed for general applications [35,36], and in particular, they have been applied to plasma-physics-related problems (see [37]). For our purposes, we chose a second order implicit symplectic algorithm inspired by Channel and Scovel.…”

“…Numerical analysis by means of a symplectic algorithm [34], implemented in a ray-tracing code, support the analytical considerations. It is worth remarking here that accurate symplectic algorithms have been recently implemented (see, for example, [35,36]), and they have been applied to the study of charged particles' motion in a plasma (see [37]). …”

Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform time stepping

“…This method, as applied to each separate Hamiltonian system, is shown to be equivalent to the method of non-canonical generating functions developed in Ref. 35, at least when the symplectic integrators are based on canonical generating functions. The method of transformation to canonical variables introduced here for 3D flows is shown numerically to work very well for the same general non-reversible, non-SDF flow.…”

“…In Appendix D we discuss optimal time stepping, i.e., finding the correct function fðx; y; zÞ that minimizes the total error over a finite time interval. 35 This optimum time stepping is derived based on backward error analysis.…”