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

• Authors: Wai Sun Don;Alex Solomonoff
• Venue: SIAM Journal on Scientific Computing
• Year: 1997

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 $k$th 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.