Symplectic Runge-Kutta Method Based Numerical Solution for the Hamiltonian Model of Spacecraft Relative Motion

Zhang, Run-de; Lee, Yang; Cai, Weiwei

doi:10.1007/978-981-13-3305-7_171

Cited by 2 publications

(1 citation statement)

References 3 publications

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“…Xing [29] developed two highly precise symplectic schemes for linear structural dynamic analysis. Zhang [30] applied the symplectic Runge-Kutta method in the dynamics of spacecraft relative motion. Peng [31,32] proposed the symplectic nonsmooth dynamic method for multibody system analysis.…”

Section: Introductionmentioning

confidence: 99%

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

sci¹

2021

AAMM

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In this paper, a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations, which mainly originate from parameters dispersions and measurement errors. Taking the perturbations into account, the perturbed system is regarded as a modification of the nominal system. By combining the perturbation series expansion method with the deterministic linear Hamiltonian system, the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived. Eventually, the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes. The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system. The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm. Numerical examples illustrate the superiority of the proposed method in accuracy and stability, especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.

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Section: Introductionmentioning

confidence: 99%

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

sci¹

2021

AAMM

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Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order

Oh,

Choe,

Kim

2024

ENGMATH

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The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. Since Hamiltonian systems have good properties such as symplecticity, numerical methods that preserve these properties have attracted the great attention. In fact, the explicit Runge-Kutta methods have used due to that schemes are very simple and its computational amounts are very small. However, the explicit schemes aren’t stable so the implicit Runge-Kutta methods have widely studied. Among those implicit schemes, symplectic numerical methods were interested. It is because it has preserved the original property of the systems. So, study of the symplectic Runge-Kutta methods have performed. The typical symplectic Runge-Kutta method is the Gauss-Legendre method, whose drawback is that it is a general implicit scheme and is too computationally expensive. Despite these drawbacks, the study of the diagonally implicit symplectic Runge-Kutta methods that preserves symplecticity has attracted much attention. The symplectic Runge-Kutta method has been studied up to sixth order in the past and efforts to obtain higher order conditions and algorithms are being intensified. In many applications such as molecular dynamics as well as in space science, such as satellite relative motion studies, this method is very effective and its application is wider. In this paper, it is presented the 7<sup>th</sup> order condition and derive the corresponding optimized method. So the diagonally implicit symplectic eleven-stages Runge-Kutta method with algebraic order 7 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

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Symplectic Runge-Kutta Method Based Numerical Solution for the Hamiltonian Model of Spacecraft Relative Motion

Cited by 2 publications

References 3 publications

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order

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