Optimal stability for trapezoidal-backward difference split-steps
The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (αΔt trapezoidal, (1 – α)Δt for BDF2) retains second-order accuracy. The ‘magic choice’ a = 2 – √2 allows the same Ja...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Oxford University Press
2013
|
| Online Access: | http://hdl.handle.net/1721.1/80879 https://orcid.org/0000-0001-7473-9287 |
| _version_ | 1826192321183481856 |
|---|---|
| author | Dharmaraja, Sohan Wang, Yinghui Strang, Gilbert |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Dharmaraja, Sohan Wang, Yinghui Strang, Gilbert |
| author_sort | Dharmaraja, Sohan |
| collection | MIT |
| description | The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (αΔt trapezoidal, (1 – α)Δt for BDF2) retains second-order accuracy. The ‘magic choice’ a = 2 – √2 allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that a = 2 – √2 also gives the largest region of linearized stability. |
| first_indexed | 2024-09-23T09:09:42Z |
| format | Article |
| id | mit-1721.1/80879 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:09:42Z |
| publishDate | 2013 |
| publisher | Oxford University Press |
| record_format | dspace |
| spelling | mit-1721.1/808792022-09-26T10:51:31Z Optimal stability for trapezoidal-backward difference split-steps Dharmaraja, Sohan Wang, Yinghui Strang, Gilbert Massachusetts Institute of Technology. Department of Mathematics Dharmaraja, Sohan Wang, Yinghui Strang, Gilbert The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (αΔt trapezoidal, (1 – α)Δt for BDF2) retains second-order accuracy. The ‘magic choice’ a = 2 – √2 allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that a = 2 – √2 also gives the largest region of linearized stability. 2013-09-23T18:29:53Z 2013-09-23T18:29:53Z 2009-09 2009-05 Article http://purl.org/eprint/type/JournalArticle 0272-4979 1464-3642 http://hdl.handle.net/1721.1/80879 Dharmaraja, S., Y. Wang, and G. Strang. “Optimal stability for trapezoidal-backward difference split-steps.” IMA Journal of Numerical Analysis 30, no. 1 (January 21, 2010): 141-148. https://orcid.org/0000-0001-7473-9287 en_US http://dx.doi.org/10.1093/imanum/drp022 IMA Journal of Numerical Analysis Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Oxford University Press MIT web domain |
| spellingShingle | Dharmaraja, Sohan Wang, Yinghui Strang, Gilbert Optimal stability for trapezoidal-backward difference split-steps |
| title | Optimal stability for trapezoidal-backward difference split-steps |
| title_full | Optimal stability for trapezoidal-backward difference split-steps |
| title_fullStr | Optimal stability for trapezoidal-backward difference split-steps |
| title_full_unstemmed | Optimal stability for trapezoidal-backward difference split-steps |
| title_short | Optimal stability for trapezoidal-backward difference split-steps |
| title_sort | optimal stability for trapezoidal backward difference split steps |
| url | http://hdl.handle.net/1721.1/80879 https://orcid.org/0000-0001-7473-9287 |
| work_keys_str_mv | AT dharmarajasohan optimalstabilityfortrapezoidalbackwarddifferencesplitsteps AT wangyinghui optimalstabilityfortrapezoidalbackwarddifferencesplitsteps AT stranggilbert optimalstabilityfortrapezoidalbackwarddifferencesplitsteps |