A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems

LIU Xiaomei; ZHOU Gang; ZHU Shuai

doi:10.21656/1000-0887.390249

Volume 40 Issue 6

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LIU Xiaomei, ZHOU Gang, ZHU Shuai. A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems[J]. Applied Mathematics and Mechanics, 2019, 40(6): 595-608. doi: 10.21656/1000-0887.390249

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A Highly Precise Symplectic Direct Integration Method Based on Phase Errors for Hamiltonian Systems

doi: 10.21656/1000-0887.390249

¹School of Science, College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209, P.R.China;²School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R.China;³School of Mechanical Engeering, Shanghai Jiao Tong University, Shanghai 200240, P.R.China

Funds: The National Natural Science Foundation of China（50876066）

Received Date: 2018-09-21
Rev Recd Date: 2018-10-11
Publish Date: 2019-06-01

Abstract

Abstract

Symplectic methods, including the generating function method, the symplectic Runge-Kutta (RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10^-13. For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.
- symplectic algorithm,
- phase error,
- structure preservation,
- Hamiltonian system,
- highly precise direct integration method

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References(22)

References

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