Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique

Abstract

A new method is investigated to reduce the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and Tal-Ezer, and the proper choice of the parameter $\alpha$, the roundoff error of the kth derivative can be reduced from $O(N^{2k})$ to $O((N \lge)^k)$, where $\epsilon$ is the machine precision and N is the number of collocation points. This drastic reduction of roundoff error makes mapped Chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large N. Several other aspects of the mapped Chebyshev differentiation matrix are also studied, revealing that the mapped Chebyshev methods require much less than $\pi$ points to resolve a wave; the eigenvalues are less sensitive to perturbation by roundoff error; and larger time steps can be used for solving PDEs. All these advantages of the mapped Chebyshev methods can be achieved while maintaining spectral accuracy.