Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 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: 10.1051/m2an/2013080
Classification: 65M06, 65N06, 65P10
Mots-clés : discrete gradient method, energy-preserving integrator, finite difference method, lagrangian mechanics
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     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},
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Yaguchi, Takaharu. Lagrangian approach to deriving energy-preserving numerical schemes for the Euler-Lagrange partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 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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