ABSTRACT
Symplectic algorithms are widely used for long-term integration of astrophysical problems. However, this technique can only be easily constructed for separable Hamiltonian, as preserving the phase-space structure. Recently, for inseparable Hamiltonian, the fourth-order extended phase-space explicit symplectic-like methods have been developed by using the Yoshida’s triple product with a mid-point map, where the algorithm is more effective, stable and also more accurate, compared with the sequent permutations of momenta and position coordinates, especially for some chaotic case. However, it has been found that, for the cases such as with chaotic orbits of spinning compact binary or circular restricted three-body system, it may cause secular drift in energy error and even more the computation break down. To solve this problem, we have made further improvement on the mid-point map with a momentum-scaling correction, which turns out to behave more stably in long-term evolution and have smaller energy error than previous methods. In particular, it could obtain a comparable phase-space distance as computing from the eighth-order Runge–Kutta method with the same time-step.
Gravitational Restricted Three-Body Problem: Existence of Retrograde Satellites at Large Distance