On an asymptotic method for computing the modified energy       for symplectic methods

Per Christian Moan; Jitse Niesen

doi:10.3934/dcds.2014.34.1105

Article Contents

2014, Volume 34, Issue 3: 1105-1120. Doi: 10.3934/dcds.2014.34.1105

This issue Previous Article Discrete gradient methods have an energy conservation law Next Article Integrability of nonholonomically coupled oscillators

On an asymptotic method for computing the modified energy for symplectic methods

Per Christian Moan^1, and
Jitse Niesen^2,

1.
Centre of Mathematics for Applications, University of Oslo
2.
School of Mathematics, University of Leeds, Leeds, LS2 9JT

Received: January 2013

Revised: April 2013

Published: March 2014

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Abstract

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory.

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Mathematics Subject Classification: Primary: 65P10; Secondary: 37J40, 37M15.

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