On Well-Conditioned Spectral Collocation and Spectral Methods by the Integral Reformulation

Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T. A. Driscoll, J. Comput. Phys., 229 (2010), pp. 5980-5998] and [L.-L. Wang, M. D. Samson, and X. Zhao, SIAM J. Sci. Comput., 36 (2014), pp. A907--A929], and the ultraspherical spectral method proposed in [S. Olver and A. Townsend, SIAM Rev., 55 (2013), pp. 462--489] for an mth-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a compact perturbation of the identity. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.