Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations Original Research Article

We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinsonʹs equation. The boundedness of waveform relaxation iterations is proved for the Hutchinsonʹs equation. This result is used in the proof of the superlinear convergence of the iterations.