ACCURACY ENHANCEMENT FOR HIGHER DERIVATIVES USING CHEBYSHEV COLLOCATION AND A MAPPING TECHNIQUE

A new method is investigated to reduce the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapp ing derived by Kosloff and Tal-Ezer, and the proper choice of the para meter al the roundoff error of the kth derivative can be reduced from O(N-2k) to O((N\ln epsilon\)(k)), where epsilon is the machine precisi on and N is the number of collocation points. This drastic reduction o f roundoff error makes mapped Chebyshev methods competitive with any o ther algorithm in computing second or higher derivatives with large N. Several other aspects of the mapped Chebyshev differentiation matrix are also studied, revealing that 1. the mapped Chebyshev methods requi re much less than pi points to resolve a wave; 2. the eigenvalues are less sensitive to perturbation by roundoff error; and 3. larger time s teps can be used for solving PDEs. All these advantages of the mapped Chebyshev methods can be achieved while maintaining spectral accuracy.