Time-symmetric integration in astrophysics
Calculating the long-term solution of ordinary differential equations, such as those of the N-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers re...
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| Format: | Article |
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Oxford University Press (OUP)
2019
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| Online Access: | http://hdl.handle.net/1721.1/121023 https://orcid.org/0000-0001-7648-0926 https://orcid.org/0000-0003-2480-5973 |
| _version_ | 1826213602571321344 |
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| author | Hernandez, David Michael Bertschinger, Edmund |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Hernandez, David Michael Bertschinger, Edmund |
| author_sort | Hernandez, David Michael |
| collection | MIT |
| description | Calculating the long-term solution of ordinary differential equations, such as those of the N-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods that are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty in accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time-stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show that it gives secular energy error increase for a pendulum problem and for a Hénon-Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favoured. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator. Keywords: methods: numerical, celestial mechanics, planets and satellites: dynamical evolution and stability, globular clusters: general |
| first_indexed | 2024-09-23T15:51:52Z |
| format | Article |
| id | mit-1721.1/121023 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T15:51:52Z |
| publishDate | 2019 |
| publisher | Oxford University Press (OUP) |
| record_format | dspace |
| spelling | mit-1721.1/1210232022-10-02T04:40:52Z Time-symmetric integration in astrophysics Hernandez, David Michael Bertschinger, Edmund Massachusetts Institute of Technology. Department of Physics MIT Kavli Institute for Astrophysics and Space Research Hernandez, David Michael Bertschinger, Edmund W Calculating the long-term solution of ordinary differential equations, such as those of the N-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods that are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty in accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time-stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show that it gives secular energy error increase for a pendulum problem and for a Hénon-Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favoured. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator. Keywords: methods: numerical, celestial mechanics, planets and satellites: dynamical evolution and stability, globular clusters: general 2019-03-18T17:58:03Z 2019-03-18T17:58:03Z 2018-04 2019-03-07T19:09:21Z Article http://purl.org/eprint/type/JournalArticle 0035-8711 1365-2966 http://hdl.handle.net/1721.1/121023 Hernandez, David M, and Edmund Bertschinger. “Time-Symmetric Integration in Astrophysics.” Monthly Notices of the Royal Astronomical Society 475, no. 4 (January 24, 2018): 5570–5584. © 2017 The Authors https://orcid.org/0000-0001-7648-0926 https://orcid.org/0000-0003-2480-5973 http://dx.doi.org/10.1093/MNRAS/STY184 Monthly Notices of the Royal Astronomical Society Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press (OUP) arXiv |
| spellingShingle | Hernandez, David Michael Bertschinger, Edmund Time-symmetric integration in astrophysics |
| title | Time-symmetric integration in astrophysics |
| title_full | Time-symmetric integration in astrophysics |
| title_fullStr | Time-symmetric integration in astrophysics |
| title_full_unstemmed | Time-symmetric integration in astrophysics |
| title_short | Time-symmetric integration in astrophysics |
| title_sort | time symmetric integration in astrophysics |
| url | http://hdl.handle.net/1721.1/121023 https://orcid.org/0000-0001-7648-0926 https://orcid.org/0000-0003-2480-5973 |
| work_keys_str_mv | AT hernandezdavidmichael timesymmetricintegrationinastrophysics AT bertschingeredmund timesymmetricintegrationinastrophysics |