Comparison between collocation methods and spectral element approach for the stability of periodic delay systems ?

This paper compares two methods that are commonly used to study the stability of delay systems. The first is a collocation technique while the second is a spectral element approach which uses the weighted residual method. Two distributions of the collocation points are compared: the first uses the extrema of Chebyshev polynomials of the first kind whereas the second uses the Legendre-Gauss-Lobatto points. The spectral element approach uses the Legendre-Gauss-Lobatto points and higher-order trial functions to discretize the delay equations while Gauss quadrature rules are used to evaluate the resulting weighted residual integrals. Two case studies are used to compare the different methods. The first case study is a 3rd order autonomous DDE while the second is a DDE describing the midspan deflections of an unbalanced rotating shaft with feedback gain (nonautonomous DDE). Convergence plots that compare the different rates of convergence of the described methods are also provided.