{"abstract":[{"text":"In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized\r\npartial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.","lang":"eng"}],"title":"Backward error analysis for variational discretisations of partial differential equations","publication_status":"published","publication":"Journal of Geometric Mechanics","file":[{"creator":"coffen","relation":"main_file","file_name":"2_BlendedBEASymmPDE.pdf","date_created":"2022-06-13T09:11:38Z","file_id":"31859","title":"Backward error analysis for variational discretisations of PDEs","access_level":"open_access","file_size":1507248,"date_updated":"2022-06-13T09:11:38Z","description":"In backward error analysis, an approximate solution to an equa-\ntion is compared to the exact solution to a nearby ‘modified’ equation. In\nnumerical ordinary differential equations, the two agree up to any power of\nthe step size. If the differential equation has a geometric property then the\nmodified equation may share it. In this way, known properties of differential\nequations can be applied to the approximation. But for partial differential\nequations, the known modified equations are of higher order, limiting appli-\ncability of the theory. Therefore, we study symmetric solutions of discretized\npartial differential equations that arise from a discrete variational principle.\nThese symmetric solutions obey infinite-dimensional functional equations. We\nshow that these equations admit second-order modified equations which are\nHamiltonian and also possess first-order Lagrangians in modified coordinates.\nThe modified equation and its associated structures are computed explicitly\nfor the case of rotating travelling waves in the nonlinear wave equation.","content_type":"application/pdf"}],"ddc":["510"],"department":[{"_id":"636"}],"volume":14,"author":[{"full_name":"McLachlan, Robert I","first_name":"Robert I","last_name":"McLachlan"},{"first_name":"Christian","id":"85279","orcid":"https://orcid.org/0000-0002-5940-8057","full_name":"Offen, Christian","last_name":"Offen"}],"date_updated":"2023-08-10T08:44:55Z","year":"2022","language":[{"iso":"eng"}],"intvolume":" 14","_id":"19941","external_id":{"arxiv":["2006.14172"]},"page":"447 - 471","status":"public","file_date_updated":"2022-06-13T09:11:38Z","oa":"1","article_type":"original","issue":"3","type":"journal_article","doi":"10.3934/jgm.2022014","has_accepted_license":"1","related_material":{"link":[{"relation":"software","url":"https://github.com/Christian-Offen/multisymplectic"}]},"citation":{"apa":"McLachlan, R. I., & Offen, C. (2022). Backward error analysis for variational discretisations of partial  differential equations. Journal of Geometric Mechanics, 14(3), 447–471. https://doi.org/10.3934/jgm.2022014","chicago":"McLachlan, Robert I, and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial  Differential Equations.” Journal of Geometric Mechanics 14, no. 3 (2022): 447–71. https://doi.org/10.3934/jgm.2022014.","short":"R.I. McLachlan, C. Offen, Journal of Geometric Mechanics 14 (2022) 447–471.","ieee":"R. I. McLachlan and C. Offen, “Backward error analysis for variational discretisations of partial  differential equations,” Journal of Geometric Mechanics, vol. 14, no. 3, pp. 447–471, 2022, doi: 10.3934/jgm.2022014.","ama":"McLachlan RI, Offen C. Backward error analysis for variational discretisations of partial  differential equations. Journal of Geometric Mechanics. 2022;14(3):447-471. doi:10.3934/jgm.2022014","bibtex":"@article{McLachlan_Offen_2022, title={Backward error analysis for variational discretisations of partial  differential equations}, volume={14}, DOI={10.3934/jgm.2022014}, number={3}, journal={Journal of Geometric Mechanics}, publisher={AIMS}, author={McLachlan, Robert I and Offen, Christian}, year={2022}, pages={447–471} }","mla":"McLachlan, Robert I., and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial  Differential Equations.” Journal of Geometric Mechanics, vol. 14, no. 3, AIMS, 2022, pp. 447–71, doi:10.3934/jgm.2022014."},"user_id":"85279","publisher":"AIMS","date_created":"2020-10-06T16:33:19Z"}