[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82400-en":3,"doc-seo-82400-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82400,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Backward Error Analysis for Matrix Discretizations of 2-D Euler Equations","We introduce a formalism of Lie–Poisson reduction of Butcher series to support backward error analysis for isospectral symplectic Runge–Kutta methods. Using a corresponding forest momentum map, the approach targets Zeitlin’s matrix discretization of the 2-D Euler equations on the sphere. Exponentially small error bounds are established for the conservation of modified Hamiltonians over exponentially long time intervals, with bounds and time-scale independent of the matrix size n when steps satisfy h = O(n−1).","arXiv :2607 .09549v1 [math .NA] 10 Jul 2026  \nBACKWARD ERROR ANALYSIS FOR MATRIX DISCRETIZATIONS  \nOF 2-D EULER EQUATIONS∗  \nEUGEN BRONASCO† AND KLAS MODIN†  \nAbstract. We introduce a formalism of Lie–Poisson reduction of Butcher series. The corresponding forest momentum map allows for describing backward error analysis of isospectral symplectic Runge–Kutta methods applied to Zeitlin’s matrix discretization of the 2-D Euler equations on the sphere. Based thereon, we obtain exponentially small error bounds for the conservation of modified Hamiltonians, valid for exponentially longtime intervals. Crucially, the error bounds and the length of the time intervals are independent of the spatial discretization parameter n (the matrix size) when the time step for different n is scaled as h = O (n−1) . Our results thus extend the classical backward error analysis result for finite-dimensional Hamiltonian systems to the infinite-dimensional case of the 2-D Euler equations discretized via matrix hydrodynamics.  \nKey words. matrix hydrodynamics, backward error analysis, Butcher series, biplanar forests, Zeitlin’s model, 2-D Euler, Hamiltonian PDEs, Lie–Poisson reduction, isospectral flows, symplectic Runge–Kutta methods  \nMSC codes. 65P10, 35Q31, 37M15, 53D50, 65M99  \n1. Introduction. The success of symplectic numerical integration schemes for long-term simulations of Hamiltonian systems can conceptually be explained as follows. Given a symplectic phase space M and a Hamiltonian system on it  \n(1.1) y˙ = XH (y) 􀀀if M = R2d then XH = J −1∇H for J = 􀀔0I I􀀕 􀀁 ,  \nlet Φh : M → M denote a numerical integration method for the system (1.1) . If w˜e try to reinterpret the discrete integrator map Φh as the exact flow of a modified vector field Xh (th˜is is the basic idea of backward error analysis), then, if Φh is a symplectic map, the vector field Xh ˜must be symplectic˜. In particular, it (locally) corresponds to a modified Hamiltonian function Hh on M, namely X = X˜Hh . This means that the discrete trajectory traced out by y k+1 = Φh (yk) corresponds to the exact flow of a true Hamiltonian system. All properties shared by solutions to Hamiltonian systems are thereby shared by the numerical trajectory. More specifically, if the method is consistent and the step size is small enough, all properties shared by Hamiltonian systems nearby the system (1.1) are shared by the numerical trajectory. For example, if the system (1.1) is integrable in the Arnold–Liouville sense, KAM-theory tells us that nearby Hamiltonian system are “almost” integrable (cf. [1]), leading to near conservation of the first integrals. As another and more generic example, a˜s long as the numerical trajectory remains on a compact subset of M, we˜have that H (yk) − Hh (yk ) = O (hp ) where p is the order of the method. Consequently, since Hh is exactly preserved by the modified vector field, the original Hamiltonian H is nearly conserved.  \nThis conceptual story of backward error analysis for symplectic integrators is pleasing and the consequences drawn from it are essentially correct for a finite-dimensional phase space M. But the story is not quite right; the devil is in the details. The core issue is a deep result in the analysis of infinite-dimensional Lie groups. The set of diffeomorphisms on M form a Lie group  \n∗ Submitted to the editors July 13, 2026  \nFunding: This work was supported by the Swedish Research Council (grant number 2022-03453), the Knut and Alice Wallenberg Foundation (grant numbers KAW 2024.0229 and KAW 2023.0433), and the G¨oran Gustafsson Foundation for Research in Natural Sciences and Medicine. The computations were enabled by resources provided by Chalmers e-Commons at Chalmers.  \n†Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Sweden ([bronasco@chalmers.se](bronasco@chalmers.se), [klas.modin@chalmers.se](klas.modin@chalmers.se))  \n1  \n2 E. BRONASCO AND K. MODIN  \n(for example in the Fr´echet topolog","cbCaig0U5IkmxTbS","https://ap.wps.com/l/cbCaig0U5IkmxTbS","pdf",624060,1,36,"English","en",105,"# Introduction\n## Backward error analysis for symplectic integrators\n## Challenge in infinite-dimensional Lie groups\n## Almost modified Hamiltonians and truncation","[{\"question\":\"What new framework is introduced for backward error analysis in this work?\",\"answer\":\"The work introduces a Lie–Poisson reduction formalism of Butcher series, supported by a forest momentum map, to analyze backward error for isospectral symplectic Runge–Kutta methods.\"},{\"question\":\"Which numerical setting and PDE are analyzed?\",\"answer\":\"The analysis applies to isospectral symplectic Runge–Kutta methods used with Zeitlin’s matrix discretization of the 2-D Euler equations on the sphere.\"},{\"question\":\"How do the error bounds depend on the matrix discretization size n and the time step h?\",\"answer\":\"With time steps scaled as h = O(n−1), the derived exponentially small error bounds and the length of exponentially long valid intervals are independent of 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new framework is introduced for backward error analysis in this work?","Question",{"text":75,"@type":76},"The work introduces a Lie–Poisson reduction formalism of Butcher series, supported by a forest momentum map, to analyze backward error for isospectral symplectic Runge–Kutta methods.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which numerical setting and PDE are analyzed?",{"text":80,"@type":76},"The analysis applies to isospectral symplectic Runge–Kutta methods used with Zeitlin’s matrix discretization of the 2-D Euler equations on the sphere.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the error bounds depend on the matrix discretization size n and the time step h?",{"text":84,"@type":76},"With time steps scaled as h = O(n−1), the derived exponentially small error bounds and the length of exponentially long valid intervals are independent of 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