Semiexplicit Symplectic Integrators for Non-separable Hamiltonian Systems

Jayawardana, Buddhika; Ohsawa, Tomoki

doi:10.48550/arxiv.2111.10915

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…There are two problems. The first problem is that, although the integrators based on the idea of Tao are symplectic in the extended phase space, it is unclear how the symplecticity of the extended phase-space Hamiltonian is related to that of the original system (Jayawardana & Ohsawa 2021). The other problem is that there is no universal method to find the optimal control parameter ω.…”

Section: Introductionmentioning

confidence: 99%

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

Zhou

Zhang

Liu

et al. 2022

ApJ

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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 options. 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 the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Zhou

Zhang

Liu

et al. 2022

ApJ

View full text Add to dashboard Cite

show abstract

“…There are two problems. One problem is that, although the integrators based on the idea of Tao are symplectic in the extended phase space, it is unclear that how the symplecticity of the extended phase space Hamiltonian is related to that of the original system (Jayawardana & Ohsawa 2021). Another problem is that there is no universal method to find the optimal control parameter ω.…”

Section: Introductionmentioning

confidence: 99%

“…The optimal choice relies on only a large number of numerical tests (Wu & Wu 2018). Combining an extended phase space approach of Pihajoki and a symmetric projection method, Jayawardana & Ohsawa (2021) have more re-cently constructed a semiexplicit symplectic integrator for inseparable Hamiltonian systems. The computations of the main time evolution for two copies of the original system with mixed-up positions and momenta are explicit.…”

Section: Introductionmentioning

confidence: 99%

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

Zhou,

Zhang,

Liu

et al. 2022

Preprint

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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 the three-part splitting method is the best one of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge-Kutta and Runge-Kutta-Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The former algorithms need small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge-Kutta and Runge-Kutta-Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracies during long-term integrations due to roundoff errors. The idea finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including the Kerr type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge-Kutta and Runge-Kutta-Nyström methods are worth recommending. Unified Astronomy Thesaurus concepts: Black hole physics (159); Computational methods (1965); Computational astronomy (293); Celestial mechanics (211)

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Semiexplicit Symplectic Integrators for Non-separable Hamiltonian Systems

Cited by 2 publications

References 16 publications

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

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

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

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