Approximation properties of a mapped Chebyshev method

Abstract

We analyze a mapped Chebyshev technique to approximate derivatives recently developed by Kosloff and Tal-Ezer. The technique is based on a one-parameter family of mappings. Earlier numerical experimentation suggests that, for suitable choices of the parameter, this mapped technique presents several advantages over the standard Chebyshev technique. Among them, better accuracy than the classical Chebyshev method. We obtain error bounds for the new technique that, when particularized to the choices of the parameter suggested in the literature, show a low order of approximation, so that, asymptotically, the new method is much less accurate than the standard Chebyshev method. These error bounds are corroborated by numerical experimentation. However, the low order behaviour of the mapped technique only appears for levels of accuracy much more stringent than those in usual computations.