Application of a symplectic integrator in a non-integrable relativistic system

Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations

Coherent or exact equations of motion for a post-Newtonian Lagrangian formalism are the Euler-Lagrange equations without any terms truncated. They naturally conserve energy {and} angular momentum. Doubling the phase-space variables of positions and momenta in the coherent equations, we establish extended phase-space symplectic-like integrators with the midpoint permutations. The velocities should be solved iteratively from the algebraic equations of the momenta defined by the Lagrangian during the course of numerical integrations. It is shown numerically that a fourth-order extended phase-spac…

Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes

In previous papers, explicit symplectic integrators were designed for nonrotating black holes, such as a Schwarzschild black hole. However, they fail to work in the Kerr spacetime because not all variables can be separable, or not all splitting parts have analytical solutions as explicit functions of proper time. To cope with this difficulty, we introduce a time transformation function to the Hamiltonian of Kerr geometry so as to obtain a time-transformed Hamiltonian consisting of five splitting parts, whose analytical solutions are explicit functions of the new coordinate time. The chosen tim…

A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes

In recent publications, the construction of explicit symplectic integrators for Schwarzschild and Kerr type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various choices. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that …

Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes

In a previous paper, second- and fourth-order explicit symplectic integrators were designed for a Hamiltonian of the Schwarzschild black hole. Following this work, we continue to trace the possibility of the construction of explicit symplectic integrators for a Hamiltonian of charged particles moving around a Reissner-Nordstrom black hole with an external magnetic field. Such explicit symplectic methods are still available when the Hamiltonian is separated into five independently integrable parts with analytical solutions as explicit functions of proper time. Numerical tests show that the prop…

后牛顿哈密顿系统的高阶显式辛分裂方法

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

提出一种新的扩展和分裂方法，用于后牛顿哈密顿系统，构建无阶降的高阶显式辛积分器，数值实验表明其长期守恒性和计算效率优于隐式方法和现有显式分裂技术。

Construction of explicit symplectic integrators in general relativity. III. Reissner-Nordstrom-(anti)-de Sitter black holes

We give a possible splitting method to a Hamiltonian for the description of charged particles moving around the Reissner-Nordstrom-(anti)-de Sitter black hole with an external magnetic field. This Hamiltonian can be separated into six analytical solvable pieces, whose solutions are explicit functions of proper time. In this case, second- and fourth-order explicit symplectic integrators are easily available. They exhibit excellent long-term behavior in maintaining the boundness of Hamiltonian errors regardless of ordered or chaotic orbits if appropriate step-sizes are chosen. Under some circums…

Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole

In a recent work of Wu, Wang, Sun and Liu, a second-order explicit symplectic integrator was proposed for the integrable Kerr spacetime geometry. It is still suited for simulating the nonintegrable dynamics of charged particles moving around the Kerr black hole embedded in an external magnetic field. Its successful construction is due to the contribution of a time transformation. The algorithm exhibits a good long-term numerical performance in stable Hamiltonian errors and computational efficiency. As its application, the dynamics of order and chaos of charged particles is surveyed. In some ci…

弯曲时空中的自适应时间步长显式辛积分器

Explicit symplectic integrators with adaptive time steps in curved spacetimes

针对弯曲时空时间变换显式辛积分器难以实现自适应时间步长的问题，团队基于Preto & Saha的步长控制技术开发出自适应版本，其计算成本低、实现简单，能提升模拟效率，可用于研究黑洞混沌动力学等场景。