Symplectic Integration without Roundoff Error

Abstract

Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious damping or excitation. Methods that do preserve all the Hamiltonian properties, i.e., for which the time-forward map is symplectic, are called symplectic integration algorithms or SIAs. Although such integrators are symplectic in theory, they are not symplectic if implemented using finite-precision arithmetic. This paper explains how to eliminate this problem by using ``Lattice SIAs'' and shows that these methods yield significant advantages when the computational error is dominated by roundoff. Using a lattice SIA is equivalent to evolving the exact solution of a problem with a Hamiltonian that is slightly different from the original. Lattice methods are useful for studies of the long-term evolution of Hamiltonian dynamical systems.