Nonlinear Stability Analysis of Area-Preserving Integrators

Linear stability analysis is inadequate for integrators designed for nondissipative systems such as Hamiltonian systems in which nonlinear effects are often decisive. Mathematical theory exists (KAM theory) for rigorous analysis of small perturbations from equilibria, but it needs to be expressed in a form that is more easily applicable to the study of area-preserving maps. We have pursued this, obtaining a completely rigorous nonlinear stability analysis for elliptic equilibria based on the Moser twist theorem and a result of Cabral and Meyer [Nonlinearity, 12 (1999), pp. 1351–1362], together with the theory of normal forms for Hamiltonian systems. The result is a determination of necessary and sufficient conditions for stability. These conditions are sharpened for the case of reversible maps and applied to the symplectic members of the Newmark family of integrators, which includes the leapfrog, the implicit midpoint, and the Störmer–Cowell methods. Nonlinear stability limits are more severe than those of linear theory. As an example, the leapfrog scheme actually has a step-size limitation of 71% of that predicted by linear analysis.