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

Published online by Cambridge University Press:  01 August 2013

Takaharu Yaguchi
Takaharu Yaguchi*
Affiliation:
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan
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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.

Keywords

Discrete gradient methodenergy-preserving integratorfinite difference methodLagrangian mechanics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1493 - 1513
Copyright
© EDP Sciences, SMAI, 2013

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