Long-term conservation of adiabatic invariants by using symplectic integrators

The advantage of symplectic integrators compared with non-symplectic Runge-Kutta integrators is shown in the computation of adiabatic invariants. Symplectic integrators do not produce a secular growth of the error, contrary to non-symplectic Runge-Kutta integrators. Furthermore, we compared the performance of 2nd, 4th, and 6th order symplectic integrators and obtained the numerical results that show SI6 has the best performance when the numerical error in the adiabatic invariants must be below 0.03%. The model systems that we used are a one-dimensional harmonic oscillator with a slowly varying frequency and a one-particle system in a slowly varying isochrone potential.