Modified trigonometric integrators

TL;DR

This paper analyzes modified trigonometric integrators for oscillatory Hamiltonian systems, highlighting the IMEX method's unique properties such as symplecticity, energy conservation, and accuracy, supported by numerical experiments.

Contribution

It identifies the IMEX method as the optimal modified trigonometric integrator with six key properties, advancing numerical integration techniques for oscillatory systems.

Findings

IMEX method is symplectic and resonance-free

It accurately captures slow energy exchange

It conserves total and oscillatory energy up to second order

Abstract

We study modified trigonometric integrators, which generalize the popular class of trigonometric integrators for highly oscillatory Hamiltonian systems by allowing the fast frequencies to be modified. Among all methods of this class, we show that the IMEX (implicit-explicit) method, which is equivalent to applying the midpoint rule to the fast, linear part of the system and the leapfrog (St\"ormer/Verlet) method to the slow, nonlinear part, is distinguished by the following properties: (i) it is symplectic; (ii) it is free of artificial resonances; (iii) it is the unique method that correctly captures slow energy exchange to leading order; (iv) it conserves the total energy and a modified oscillatory energy up to to second order; (v) it is uniformly second-order accurate in the slow components; and (vi) it has the correct magnitude of deviations of the fast oscillatory energy, which is an adiabatic invariant. These theoretical results are supported by numerical experiments on the Fermi-Pasta-Ulam problem and indicate that the IMEX method, for these six properties, dominates the class of modified trigonometric integrators.