Discrete gradient methods have an energy conservation law

Robert I. McLachlan; G. R. W. Quispel

doi:10.3934/dcds.2014.34.1099

Article Contents

2014, Volume 34, Issue 3: 1099-1104. Doi: 10.3934/dcds.2014.34.1099

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Discrete gradient methods have an energy conservation law

Robert I. McLachlan^1, and
G. R. W. Quispel^2,

1.
Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North
2.
Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086

Received: January 2013

Revised: April 2013

Published: March 2014

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Abstract

We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved density as the continuous conservation law, while its flux is found by replacing all derivatives of the conserved density appearing in the continuous flux by discrete gradients.

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Mathematics Subject Classification: Primary: 35L65, 65M20; Secondary: 65P10.

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