Numerical discretization of completely integrable nonlinear Hamiltonian systems

Completely integrable nonlinear Hamiltonian systems offer attractive models for novel accelerator systems that are being developed to push the frontiers of high intensity and high power beam applications. However, all known numerical methods used to discretize and simulate these systems spoil the integrability property of the original, time-continuous, theoretically integrable system. Since the hallmark of integrability is the existence of sufficiently many functionally independent first integrals in involution, we investigated several methods’ performance in terms of invariant preservation over longer timescales. We conclude that, while novel structure preserving integrators would be ideal, symplectic methods under mild restrictions offer adequate invariant preservation. We especially highlight the very good performance of the Stormer-Verlet method at low accuracy and a modified Lobatto method at high accuracy for the Integrable Optics Test Accelerator (IOTA). However, we show that symplectic integrators are limited by numerically induced resonances, which make interpretation of some simulation results challenging. These limitations and workarounds are discussed.

Abstract licence: CC BY

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