Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method

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, α-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 ≲ 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 α (Hesthaven, Dinesen, and Lynov [{\em J.\ Comput.\ Phys}., 155 (1999), pp. 287--306]) and for α scaled with N , α sufficiently different from 1 (Don and Solomonoff [ SIAM J. Sci. Comput. , 18 (1997), pp. 1040–1055]). Theoretical bounds, however, show that any realistic choice for α, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step fora stable explicit integrator in time, much less than the O ( N ) improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing α carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than π 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 α 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.