Multi-Point Hermite Methods for the N-Body Problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916579816505344 |
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| author | Dittmann, Alexander J. |
| author_facet | Dittmann, Alexander J. |
| contents | Numerical integration methods are central to the study of self-gravitating systems, particularly those comprised of many bodies or otherwise beyond the reach of analytical methods. Predictor-corrector schemes, both multi-step methods and those based on 2-point Hermite interpolation, have found great success in the simulation of star clusters and other collisional systems. Higher-order methods, such as those based on Gaussian quadratures and Richardson extrapolation, have also proven popular for high-accuracy integrations of few-body systems, particularly those that may undergo close encounters. This work presents a family of high-order schemes based on multi-point Hermite interpolation. When applied as a multi-step multi-derivative schemes, these can be seen as generalizing both Adams-Bashforth-Moulton methods and 2-point Hermite methods; I present results for the 6th-, 9th-, and 12th-order 3-point schemes applied in this manner using variable time steps. In a cluster-like test problem, the 3-point 6th-order predictor-corrector scheme matches or outperforms the standard 2-point 4th-order Hermite scheme at negligible O(N) cost. I also present a number of high-order time-symmetric schemes up to 18th order, which have the potential to improve the accuracy and efficiency of long-duration simulations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-Point Hermite Methods for the N-Body Problem Dittmann, Alexander J. Instrumentation and Methods for Astrophysics Earth and Planetary Astrophysics Astrophysics of Galaxies Numerical Analysis Computational Physics Numerical integration methods are central to the study of self-gravitating systems, particularly those comprised of many bodies or otherwise beyond the reach of analytical methods. Predictor-corrector schemes, both multi-step methods and those based on 2-point Hermite interpolation, have found great success in the simulation of star clusters and other collisional systems. Higher-order methods, such as those based on Gaussian quadratures and Richardson extrapolation, have also proven popular for high-accuracy integrations of few-body systems, particularly those that may undergo close encounters. This work presents a family of high-order schemes based on multi-point Hermite interpolation. When applied as a multi-step multi-derivative schemes, these can be seen as generalizing both Adams-Bashforth-Moulton methods and 2-point Hermite methods; I present results for the 6th-, 9th-, and 12th-order 3-point schemes applied in this manner using variable time steps. In a cluster-like test problem, the 3-point 6th-order predictor-corrector scheme matches or outperforms the standard 2-point 4th-order Hermite scheme at negligible O(N) cost. I also present a number of high-order time-symmetric schemes up to 18th order, which have the potential to improve the accuracy and efficiency of long-duration simulations. |
| title | Multi-Point Hermite Methods for the N-Body Problem |
| topic | Instrumentation and Methods for Astrophysics Earth and Planetary Astrophysics Astrophysics of Galaxies Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2410.17311 |