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Bespoke finite difference schemes that preserve multiple conservation laws

Part of: Numerical problems in dynamical systems Infinite-dimensional Hamiltonian systems Computational aspects of field theory and polynomials Difference and functional equations Difference equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 May 2015

Timothy J. Grant
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.

MSC classification

Primary: 39A14: Partial difference equations 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 65M06: Finite difference methods 65P99: None of the above, but in this section
Secondary: 12Y99: None of the above, but in this section 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 372 - 403
Copyright
© The Author 2015 

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