Time-symmetry, symplecticity and stability of Euler-Maclaurin and Lanczos-Dyche integration
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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
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2019
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| _version_ | 1866913619632979968 |
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| author | Markakis, Charalampos M. O'Boyle, Michael F. Glennon, Derek Tran, Khoa Brubeck, Pablo Haas, Roland Schive, Hsi-Yu Uryū, Kōji |
| author_facet | Markakis, Charalampos M. O'Boyle, Michael F. Glennon, Derek Tran, Khoa Brubeck, Pablo Haas, Roland Schive, Hsi-Yu Uryū, Kōji |
| contents | Numerical evolution of time-dependent differential equations via explicit Runge-Kutta or Taylor methods typically fails to preserve symmetries of a system. It is known that there exists no numerical integration method that in general preserves both the energy and the symplectic structure of a Hamiltonian system. One is thus normally forced to make a choice. Nevertheless, a symmetric integration formula, obtained by Lanczos-Dyche via two-point Taylor expansion (or Hermite interpolation), is shown here to preserve both energy as well as symplectic structure for linear systems. This formula shares similarities with the Euler-Maclaurin formula, but is superconvergent rather than asymptotically convergent. For partial differential equations, the resulting evolution methods are unconditionally stable, i.e, not subject to a Courant-Friedrichs-Lewy limit. Although generally implicit, these methods become explicit for linear systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_09967 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Time-symmetry, symplecticity and stability of Euler-Maclaurin and Lanczos-Dyche integration Markakis, Charalampos M. O'Boyle, Michael F. Glennon, Derek Tran, Khoa Brubeck, Pablo Haas, Roland Schive, Hsi-Yu Uryū, Kōji Numerical Analysis Mathematical Physics Computational Physics 65P10 (Primary) 65L20, 65M20 (Secondary) Numerical evolution of time-dependent differential equations via explicit Runge-Kutta or Taylor methods typically fails to preserve symmetries of a system. It is known that there exists no numerical integration method that in general preserves both the energy and the symplectic structure of a Hamiltonian system. One is thus normally forced to make a choice. Nevertheless, a symmetric integration formula, obtained by Lanczos-Dyche via two-point Taylor expansion (or Hermite interpolation), is shown here to preserve both energy as well as symplectic structure for linear systems. This formula shares similarities with the Euler-Maclaurin formula, but is superconvergent rather than asymptotically convergent. For partial differential equations, the resulting evolution methods are unconditionally stable, i.e, not subject to a Courant-Friedrichs-Lewy limit. Although generally implicit, these methods become explicit for linear systems. |
| title | Time-symmetry, symplecticity and stability of Euler-Maclaurin and Lanczos-Dyche integration |
| topic | Numerical Analysis Mathematical Physics Computational Physics 65P10 (Primary) 65L20, 65M20 (Secondary) |
| url | https://arxiv.org/abs/1901.09967 |