Generalized flow-composed symplectic methods for post-Newtonian Hamiltonian systems

TL;DR

This paper introduces a generalized flow-composed Runge--Kutta method tailored for nonseparable post-Newtonian Hamiltonian systems, demonstrating improved accuracy and efficiency over existing methods through numerical experiments.

Contribution

It develops a new symplectic integrator for nonseparable Hamiltonian systems with a specific formulation, enhancing numerical performance in modeling compact binaries.

Findings

The GFCRK method is symplectic when the underlying RK method is symplectic.

Numerical experiments show higher accuracy of GFCRK in weak PN effects.

GFCRK outperforms semi-explicit mixed symplectic methods in efficiency.

Abstract

Due to the nonseparability of the post-Newtonian (PN) Hamiltonian systems of compact objects, the symplectic methods that admit the linear error growth and the near preservation of first integrals are always implicit as explicit symplectic methods have not been currently found for general nonseparable Hamiltonian systems. Since the PN Hamiltonian has a particular formulation that includes a dominant Newtonian part and a perturbation PN part, we present the generalized flow-composed Runge--Kutta (GFCRK) method with a free parameter $\lambda$ to PN Hamiltonian systems. It is shown that the GFCRK method is symplectic once the underlying RK method is symplectic, and it is symmetric once the underlying RK method is symmetric under the setting $\lambda=1/2$. Numerical experiments with the 2PN Hamiltonian of spinning compact binaries demonstrate the higher accuracy and efficiency of the symplectic GFCRK method than the underlying symplectic RK method in the case of weak PN effect. Meanwhile, the numerical results also support higher efficiency of the symplectic GFCRK method than the semi-explicit mixed symplectic method of the same order.