On the Numerical Stability of Some Symplectic Integrators

In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear.