High Frequency Induced Instability in Nyström Methods for the van der Pol Equation

In this paper several Nyström 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 Nyström, 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 Nyström 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.