Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 5, p. 1493-1513

We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler-Lagrange partial differential equations. Noether's theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler-Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter part of this statement and the discrete gradient method. It is also shown that the symmetry of space translation derives momentum-preserving schemes. Finally, we discuss the existence of discrete local conservation laws.

DOI : https://doi.org/10.1051/m2an/2013080
Classification:  65M06,  65N06,  65P10
Keywords: discrete gradient method, energy-preserving integrator, finite difference method, lagrangian mechanics
@article{M2AN_2013__47_5_1493_0,
     author = {Yaguchi, Takaharu},
     title = {Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     pages = {1493-1513},
     doi = {10.1051/m2an/2013080},
     zbl = {1284.65109},
     mrnumber = {3100772},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_5_1493_0}
}
Yaguchi, Takaharu. Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 5, pp. 1493-1513. doi : 10.1051/m2an/2013080. http://www.numdam.org/item/M2AN_2013__47_5_1493_0/

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