Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations

The waveform relaxation technique for linear parabolic differential and dif ferential-functional equations is studied. We use the second order finite d ifference method and the Chebyshev pseudospectral method for spatial discre tization and apply a Gauss-Seidel waveform relaxation scheme to the resulti ng systems of ordinary differential and differential-functional equations. Waveform relaxation error bounds are presented for the two semi-discretizat ion schemes in both functional and non-functional cases. Sharp error bounds are obtained after application of an inequalifies technique with time-depe ndent coefficients and logarithmic norm. Convergence of the schemes is stud ied analytically and compared by means of extensive numerical data obtained for four parabolic equations with different coefficients. Our conclusion i s that waveform relaxation error bounds and waveform relaxation convergence are better after Chebyshev pseudospectral semi-discretization than after f inite difference method. Moreover, for the same accuracy of semi-discretiza tion, a single Gauss-Seidel waveform iteration after Chebyshev pseudospectr al method is less expensive than after finite difference method. (C) 2000 I MACS. Published by Elsevier Science B.V. All rights reserved.