High order compact finite difference schemes on nonuniform grids

Abstract

In this paper we consider high-order compact finite difference schemes constructed on 1D non-uniform grids. We apply them to parabolic and Schr{\" o}\-din\-ger equations. Stability of these schemes is investigated by using the spectral method. Computer experiments are applied in order to find critical grids for which the stability condition is violated. Such grids are obtained for the Schr{\" o}dinger problem, but not for the parabolic problems. Numerical examples supporting our theoretical analysis are provided and discussed.