High-Order, Implicit Time Integration of Discrete, Chaotic Dynamical Systems
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911744769654784 |
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| author | Morozova, Viktoriya Coder, James G. Holst, Kevin |
| author_facet | Morozova, Viktoriya Coder, James G. Holst, Kevin |
| contents | A wide range of implicit time integration methods, including multi-step, implicit Runge-Kutta, and Galerkin finite-time element schemes, is evaluated in the context of chaotic dynamical systems. The schemes are applied to solve the Lorenz equations, the equation of motion of a Duffing oscillator, and the Kuramoto-Sivashinsky system, with the goal of finding the most computationally efficient method that results in the least expensive model for a chosen level of accuracy. It is found that the quasi-period of a chaotic system strongly limits the time-step size that can be used in the simulations, and all schemes fail once the time-step size reaches a significant fraction of that period. In these conditions, the computational cost per time-step becomes one of the most important factors determining the efficiency of the schemes. The cheaper, second-order schemes are shown to have an advantage over the higher-order schemes at large time-step sizes, with one possible exception being the fourth-order continuous Galerkin scheme. The higher-order schemes become more efficient than the lower-order schemes as accuracy requirements tighten. If going beyond the second-order is necessary for reasons other than computational efficiency, the fourth-order methods are shown to perform better than the third-order ones at all time-step sizes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00576 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | High-Order, Implicit Time Integration of Discrete, Chaotic Dynamical Systems Morozova, Viktoriya Coder, James G. Holst, Kevin Computational Physics A wide range of implicit time integration methods, including multi-step, implicit Runge-Kutta, and Galerkin finite-time element schemes, is evaluated in the context of chaotic dynamical systems. The schemes are applied to solve the Lorenz equations, the equation of motion of a Duffing oscillator, and the Kuramoto-Sivashinsky system, with the goal of finding the most computationally efficient method that results in the least expensive model for a chosen level of accuracy. It is found that the quasi-period of a chaotic system strongly limits the time-step size that can be used in the simulations, and all schemes fail once the time-step size reaches a significant fraction of that period. In these conditions, the computational cost per time-step becomes one of the most important factors determining the efficiency of the schemes. The cheaper, second-order schemes are shown to have an advantage over the higher-order schemes at large time-step sizes, with one possible exception being the fourth-order continuous Galerkin scheme. The higher-order schemes become more efficient than the lower-order schemes as accuracy requirements tighten. If going beyond the second-order is necessary for reasons other than computational efficiency, the fourth-order methods are shown to perform better than the third-order ones at all time-step sizes. |
| title | High-Order, Implicit Time Integration of Discrete, Chaotic Dynamical Systems |
| topic | Computational Physics |
| url | https://arxiv.org/abs/2401.00576 |