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

We study a new method in reducing the roundo error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Koslo and Tal-Ezer, and the proper choice of the parameter , the roundo error of the k-th derivative can be reduced from O(N2k) to O((N jln j)k), where is the machine precision and N is the number of collocation points. This drastic reduction of roundo error makes mapped Chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large N . We also study several other aspects of the mapped Chebyshev di erentiation matrix. We nd that 1) the mapped Chebyshev methods requires much less than points to resolve a wave, 2) the eigenvalues are less sensitive to perturbation by roundo error, and 3) larger time steps can be used for solving PDEs. All these advantages of the mapped Chebyshev methods can be achieved while maintaining spectral accuracy.