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.
Mots clés : 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 },
pages = {1493--1513},
publisher = {EDP-Sciences},
volume = {47},
number = {5},
year = {2013},
doi = {10.1051/m2an/2013080},
mrnumber = {3100772},
zbl = {1284.65109},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013080/}
}
TY - JOUR AU - Yaguchi, Takaharu TI - Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1493 EP - 1513 VL - 47 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013080/ DO - 10.1051/m2an/2013080 LA - en ID - M2AN_2013__47_5_1493_0 ER -
%0 Journal Article %A Yaguchi, Takaharu %T Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1493-1513 %V 47 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013080/ %R 10.1051/m2an/2013080 %G en %F 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 , Tome 47 (2013) no. 5, pp. 1493-1513. doi : 10.1051/m2an/2013080. http://www.numdam.org/articles/10.1051/m2an/2013080/
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