Accuracy, Resolution, and Stability Properties of a Modified Chebyshev Method

• Authors: Jodi L. Mead;Rosemary A. Renaut
• Venue: SIAM Journal on Scientific Computing
• Year: 2002

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order $M$ requires time steps of approximately $O(N^{-2M})$ for stable explicit solvers. Theoretically, time steps may be increased to $O(N^{-M})$ with the use of a parameter, $\alpha$-dependent mapped method introduced by Kosloff and Tal-Ezer [{\em J.\ Comput.\ Phys}., 104 (1993), pp. 457--469]. Our analysis focuses on the utilization of this method for reasonable practical choices for $N$, namely $N \lesssim 30$, as may be needed for two- or three-dimensional modeling. Results presented confirm that spectral accuracy with increasing $N$ is possible both for constant $\alpha$ (Hesthaven, Dinesen, and Lynov [{\em J.\ Comput.\ Phys}., 155 (1999), pp. 287--306]) and for $\alpha$ scaled with $N$, $\alpha$ sufficiently different from $1$ (Don and Solomonoff [{\em SIAM J.\ Sci.\ Comput}., 18 (1997), pp. 1040--1055]). Theoretical bounds, however, show that any realistic choice for $\alpha$, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step for a stable explicit integrator in time, much less than the $O(N)$ improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing $\alpha$ carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than $\pi$ points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about $8$ points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while $\alpha$ can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of $N$, it is not possible to achieve both single precision accuracy and gain the advantage of an $O(N^{-M})$ time step.