Comparison of second-order mixed symplectic integrator between semi-implicit Euler method and implicit midpoint rule

Abstract

When a Hamiltonian can be split into integrable and nonintegrable parts, the former part is solved analytically, and the latter one is integrated numerically by means of implicit symplectic integrators such as the first-order semi-implicit Euler method or the second-order implicit midpoint rule. These analytical and numerical solutions are used to construct a second-order mixed symplectic integrator with the semi-implicit Euler method and one with the implicit midpoint rule. A theoretical analysis shows that the Euler mixed integrator is inferior to the midpoint one in the sense of numerical stability. Numerical simulations of the circularly-restricted three-body problem also support this fact. It is further shown through numerical integrations of the post-Newtonian Hamiltonian of spinning compact binaries that the qualities of the Euler mixed integrator and the midpoint mixed method do depend on the type of orbits. Especially for chaotic orbits, the Euler mixed integrator often becomes unstable. In addition, the Euler mixed integrator has an advantage over the midpoint mixed method in computational efficiency, and is almost equivalent to the latter in the numerical accuracy if the two mixed integrators are stable. In spite of this, the midpoint mixed integrator is worth recommending for the study of the dynamics of post-Newtonian Hamiltonians of spinning compact binaries.