High frequency induced instability in Nyström methods for the Van der Pol equation

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dc.contributor.author Schoombie, S.W.
dc.contributor.author Mare, Eben
dc.date.accessioned 2009-01-16T06:54:08Z
dc.date.available 2009-01-16T06:54:08Z
dc.date.issued 2008
dc.description.abstract In this paper several Nystrom methods for the van der Pol equation are considered. In an earlier study by Cai, Aoyagi, and Abe it was shown that the second order Nystrom, or leapfrog, method fails to approximate the limit cycle of the van der Pol equation, exhibiting a periodic modulation of the amplitude and sporadic high frequency noise instead. Cai et al. did a linear analysis and concluded that the spurious behavior was due to the interaction of the main part of the solution with a high frequency computational mode. In this paper we also apply a third and fourth order Nystrom method to the van der Pol equation. Numerical experiments show that in these cases the high frequency mode causes blowup after some time. The onset of the instability can be delayed by decreasing the time step. We also improve on their analysis of the second order scheme by doing a nonlinear analysis, to wit a discrete multiple scales analysis. By this means we are able to explain the spurious behavior of this system completely. en_US
dc.identifier.citation Schoombie, SW & Maré, E 2008, ‘High frequency induced instability in Nyström methods for the Van der Pol equation’, SIAM Journal of Numerical Analysis, vol. 46, no. 3, pp. 1-19. [http://www.siam.org/journals/sinum.php] en_US
dc.identifier.issn 0036-1429
dc.identifier.other 10.1137/050644148
dc.identifier.uri http://hdl.handle.net/2263/8617
dc.language.iso en en_US
dc.publisher Society for Industrial and Applied Mathematics en_US
dc.rights Society for Industrial and Applied Mathematics en_US
dc.subject Van der Pol equation en_US
dc.subject Spurious solutions en_US
dc.subject Nyström en_US
dc.subject Leapfrog en_US
dc.subject Multiple scales en_US
dc.subject Discrete approximation en_US
dc.subject Diference equations en_US
dc.subject.lcsh Difference equations en
dc.subject.lcsh Equations en
dc.subject.lcsh Calculus en
dc.title High frequency induced instability in Nyström methods for the Van der Pol equation en_US
dc.type Article en_US


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