On the Pseudo-Spectral Method and Spectral Accuracy

There are numerical accuracy problems related to the implementation of sharp internal interfaces in pseudo-spectral and finite-difference schemes. It is common practice to classify numerical errors due to the implementation of interfaces as being to some order in a Taylor expansion. An alternative approach is to classify these errors as being to some order in a Fourier expansion.The pseudo-spectral method does not provide spectral accuracy in inhomogeneous media. The numerical errorsfor the upper half of the frequency/wavenumber spectra of the propagating fields are not related to theimplementation of the derivative operators but to aliasing effects coming from the multiplicationof static material-parameter fields with the dynamic, propagating, fields. The pseudo-spectral methodcan only provide half-spectral accuracy. The same type of spatial aliasing errors are present also forfinite-difference schemes. High-order finite differences can provide the same accuracy as the pseudo-spectral method if the staggered finite-difference derivative operators have a negligible errorat four grid points per shortest wavelength and above. Smoothing of the material-parameter field leads to additional reduction in the error-free bandwidthof the propagating fields. Assuming that there is a maximum wavenumber up to which the spectrumof the smoothed model coincide with the implementation using a properly bandlimited Heaviside step function, then there exists a local critical wavenumber for the propagating field equal toone half of the maximum wavenumber for the smoothed model. Harmonic averaging of material-parameter fields also results in wavenumber spectra where there is a maximum wavenumber above whichthe wavenumber spectrum deviates from an implementation with a bandlimited Heaviside step function.The same one-half rule is applicable also in this case.