Ws. Don et A. Solomonoff, ACCURACY ENHANCEMENT FOR HIGHER DERIVATIVES USING CHEBYSHEV COLLOCATION AND A MAPPING TECHNIQUE, SIAM journal on scientific computing, 18(4), 1997, pp. 1040-1055
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.