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Volume 40, Issue 4
A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

Wei Shi & Kai Liu

J. Comp. Math., 40 (2022), pp. 570-588.

Published online: 2022-04

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  • Abstract

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.

  • AMS Subject Headings

65L05, 65L07, 65L20, 65P10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

shuier628@163.com (Wei Shi)

laukai520@163.com (Kai Liu)

  • BibTex
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@Article{JCM-40-570, author = {Shi , Wei and Liu , Kai}, title = {A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {4}, pages = {570--588}, abstract = {

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2011-m2019-0272}, url = {http://global-sci.org/intro/article_detail/jcm/20501.html} }
TY - JOUR T1 - A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems AU - Shi , Wei AU - Liu , Kai JO - Journal of Computational Mathematics VL - 4 SP - 570 EP - 588 PY - 2022 DA - 2022/04 SN - 40 DO - http://doi.org/10.4208/jcm.2011-m2019-0272 UR - https://global-sci.org/intro/article_detail/jcm/20501.html KW - Weakly dissipative systems, Oscillatory systems, Structure-preserving algorithm, Discrete gradient integrator, Sine-Gordon equation, Continuous $\alpha$-Fermi-Pasta-Ulam system. AB -

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.

Wei Shi & Kai Liu. (2022). A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems. Journal of Computational Mathematics. 40 (4). 570-588. doi:10.4208/jcm.2011-m2019-0272
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