article published Robert I McLachlan author Christian Offen author 85279https://orcid.org/0000-0002-5940-8057 636 department 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 partial 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. https://ris.uni-paderborn.de/download/19941/31859/2_BlendedBEASymmPDE.pdf application/pdfno AIMS2022 eng 2006.1417210.3934/jgm.2022014 143447 - 471 https://github.com/Christian-Offen/multisymplectic R.I. McLachlan, C. Offen, Journal of Geometric Mechanics 14 (2022) 447–471. McLachlan, Robert I., and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial  Differential Equations.” <i>Journal of Geometric Mechanics</i>, vol. 14, no. 3, AIMS, 2022, pp. 447–71, doi:<a href="https://doi.org/10.3934/jgm.2022014">10.3934/jgm.2022014</a>. @article{McLachlan_Offen_2022, title={Backward error analysis for variational discretisations of partial  differential equations}, volume={14}, DOI={<a href="https://doi.org/10.3934/jgm.2022014">10.3934/jgm.2022014</a>}, number={3}, journal={Journal of Geometric Mechanics}, publisher={AIMS}, author={McLachlan, Robert I and Offen, Christian}, year={2022}, pages={447–471} } McLachlan, R. I., &#38; Offen, C. (2022). Backward error analysis for variational discretisations of partial  differential equations. <i>Journal of Geometric Mechanics</i>, <i>14</i>(3), 447–471. <a href="https://doi.org/10.3934/jgm.2022014">https://doi.org/10.3934/jgm.2022014</a> R. I. McLachlan and C. Offen, “Backward error analysis for variational discretisations of partial  differential equations,” <i>Journal of Geometric Mechanics</i>, vol. 14, no. 3, pp. 447–471, 2022, doi: <a href="https://doi.org/10.3934/jgm.2022014">10.3934/jgm.2022014</a>. McLachlan, Robert I, and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial  Differential Equations.” <i>Journal of Geometric Mechanics</i> 14, no. 3 (2022): 447–71. <a href="https://doi.org/10.3934/jgm.2022014">https://doi.org/10.3934/jgm.2022014</a>. McLachlan RI, Offen C. Backward error analysis for variational discretisations of partial  differential equations. <i>Journal of Geometric Mechanics</i>. 2022;14(3):447-471. doi:<a href="https://doi.org/10.3934/jgm.2022014">10.3934/jgm.2022014</a> 199412020-10-06T16:33:19Z2023-08-10T08:44:55Z