Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations∗
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan
Received: 27 April 2012
Revised: 12 December 2012
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.
Mathematics Subject Classification: 65M06 / 65N06 / 65P10
Key words: Discrete gradient method / energy-preserving integrator / finite difference method / Lagrangian mechanics
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