Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations

Takaharu Yaguchi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 5, page 1493-1513
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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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 - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1493-1513. <http://eudml.org/doc/273182>.

@article{Yaguchi2013,
abstract = {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.},
author = {Yaguchi, Takaharu},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {discrete gradient method; energy-preserving integrator; finite difference method; lagrangian mechanics; Lagrangian mechanics; Hamilton-Jacobi equation; conservation law},
language = {eng},
number = {5},
pages = {1493-1513},
publisher = {EDP-Sciences},
title = {Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations},
url = {http://eudml.org/doc/273182},
volume = {47},
year = {2013},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 5
SP - 1493
EP - 1513
AB - 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.
LA - eng
KW - discrete gradient method; energy-preserving integrator; finite difference method; lagrangian mechanics; Lagrangian mechanics; Hamilton-Jacobi equation; conservation law
UR - http://eudml.org/doc/273182
ER -

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