FFT vs. conjugate gradient method for solution of flow equations by pseudo-spectral methods

The speed in pseudospectral calculations of the conjugate gradient method is compared with that of the fast Fourier transform method, and a high speed, combined sine-Chebyshev technique for the solution of Poisson's equation is presented. It is found that, in general, at least four times as many terms may be required in a sine expansion to yield accuracies equivalent to those of the Chebyshev expansion. Both the sine and Chebyshev methods fit the function being represented as well as could be expected throughout the entire domain, implying that a combined Chebyshev-Fourier approach to the solution of Poisson's equation is a feasible alternative to the costly Chebyshev procedure.