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Bespoke finite difference schemes that preserve multiple conservation laws

Published online by Cambridge University Press:  01 May 2015

Timothy J. Grant*
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK email timothy_grant@hotmail.co.uk

Abstract

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Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.

Type
Research Article
Copyright
© The Author 2015 

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