{"doi":"10.3934/jgm.2023005","file":[{"file_name":"BEA_MultiStep_Matrix.pdf","file_size":827030,"date_updated":"2022-08-12T16:48:59Z","file_id":"32801","title":"Backward error analysis for conjugate symplectic methods","relation":"main_file","access_level":"open_access","content_type":"application/pdf","creator":"coffen","description":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","date_created":"2022-08-12T16:48:59Z"}],"page":"98-115","ddc":["510"],"publication_status":"published","issue":"1","keyword":["variational integrators","backward error analysis","Euler--Lagrange equations","multistep methods","conjugate symplectic methods"],"file_date_updated":"2022-08-12T16:48:59Z","related_material":{"link":[{"url":"https://github.com/Christian-Offen/BEAConjugateSymplectic","relation":"software"}]},"date_created":"2022-01-11T12:48:39Z","has_accepted_license":"1","quality_controlled":"1","oa":"1","abstract":[{"text":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","lang":"eng"}],"article_type":"original","type":"journal_article","date_updated":"2023-08-10T08:40:30Z","publication":"Journal of Geometric Mechanics","language":[{"iso":"eng"}],"_id":"29236","title":"Backward error analysis for conjugate symplectic methods","department":[{"_id":"636"}],"volume":15,"user_id":"85279","external_id":{"arxiv":["2201.03911"]},"citation":{"chicago":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>.","short":"R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115.","bibtex":"@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate symplectic methods}, volume={15}, DOI={<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>}, number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press}, author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }","apa":"McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>","mla":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS Press, 2023, pp. 98–115, doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","ama":"McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>","ieee":"R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023, doi: <a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>."},"status":"public","intvolume":"        15","year":"2023","publisher":"AIMS Press","author":[{"first_name":"Robert","full_name":"McLachlan, Robert","last_name":"McLachlan"},{"full_name":"Offen, Christian","id":"85279","last_name":"Offen","orcid":"0000-0002-5940-8057","first_name":"Christian"}]}