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, Kolmogorov-Arnold-Moser theory, and additional multiscale analysis, an error bound of O(Tδ^{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.

Efficient evaluation of arbitrary static electromagnetic fields with applications for symplectic particle tracking

“…Until recently, the general belief was that higher order symplectic integrators for general non-separable Hamiltonians have to be implicit, although for specific cases explicit integrators were available [1,22]. An explicit second-order method for general Hamiltonians, symplectic in extended phase space, is described in [23]. We implemented this secondorder explicit symplectic integrator, which has the correct long time behavior.…”

Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. II. Test Particles in Electromagnetic Fields and GRMHD

“…While explicit symplectic schemes can be constructed, they are generally not applicable to systems characterized by inseparable Hamiltonians (see Section 3.3), resulting in energy errors that are not bounded in time (although the increase in error is generally very slow, see Tao 2016). Instead, one can rely on implicit symplectic schemes, such as the implicit midpoint rule (IMR from now on), which is the simplest second-order, symplectic, implicit integration scheme (Hairer et al 2006).…”

Numerical integration in Celestial Mechanics: a case for contact geometry

“…For us the relevance of equation (34) lies in the observation that such equation belongs to the class (1) and therefore it has a natural description in terms of contact geometry. In fact, the contact Hamiltonian for the Lane-Emden equation is…”