Spectral difference methods for solving differential equations

A family of recently developed techniques is explored for achieving both matrix sparsity and rapid convergence when numerically solving differential and eigenvalue equations without domain decomposition. These methods, which we call spectral differences, include Boydʹs sum acceleration techniques and the Lagrange distributed approximating functional (LDAF) approach. A formula is developed for estimating the unknown Gaussian parameter within LDAF. We implement these methods to calculate the Morse vibrational energies for diatomic iodine. For equivalent bandwidths the sum acceleration with finite difference weights generates energies which are between two and three orders of magnitude more accurate than those from LDAF.