arXiv subjects

Lijie Mei

Publications and source records attributed to Lijie Mei.

3 recordsLinked to original sources

arXiv2026

Efficient high-order explicit symplectic splitting methods for post-Newtonian Hamiltonian systems

The nonseparability of post-Newtonian (PN) Hamiltonian systems typically necessitates the use of computationally expensive implicit integrators. Recent research overcomes this limitation by embedding the dynamics into a doubled phase space, which enables the development of explicit symplectic methods. However, existing specially designed explicit integrators suffer from order reduction for high-order methods when the time stepsize is small, i.e., $h <\varepsilon^3$. In this paper, we propose a novel extension and splitting approach for the doubled Hamiltonian, under which specially designed explicit symplectic integrators can be constructed. It is shown that the proposed integrators achieve genuine high-order convergence without order reduction and take advantage of the small PN parameter $\varepsilon$. Numerical results from simulations with 2PN spinning binaries demonstrate superior long-term conservation of invariants and significantly higher computational efficiency compared to both implicit methods and existing explicit splitting techniques.

astro-ph.IM↗

arXiv2025

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems is an open and important problem in both numerical analysis and computing in science and engineering, as explicit integrators are usually more efficient than the implicit integrators of the same order of accuracy. Up to now, all responses to this problem are negative. That is, there exist explicit symplectic integrators only for some special nonseparable Hamiltonian systems, whereas the universal design involving explicit symplectic integrators for general nonseparable Hamiltonian systems has not yet been studied sufficiently. In this paper, we present a constructive proof for the existence of explicit symplectic integrators for general nonseparable Hamiltonian systems via finding explicit symplectic mappings under which the special submanifold of the extended phase space is invariant. It turns out that the proposed explicit integrators are symplectic in both the extended phase space and the original phase space. Moreover, on the basis of the global modified Hamiltonians of the proposed integrators, the backward error analysis is made via a parameter relaxation and restriction technique to show the linear growth of global errors and the near-preservation of first integrals. In particular, the effective estimated time interval is nearly the same as classical implicit symplectic integrators when applied to (near-) integrable Hamiltonian systems. Numerical experiments with a completely integrable nonseparable Hamiltonian and a nonintegrable nonseparable Hamiltonian illustrate the good long-term behavior and high efficiency of the explicit symplectic integrators proposed and analyzed in this paper.

math.NA↗

arXiv2024

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

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.

astro-ph.IM↗