There are numerical accuracy problems related to the implementation of sharp internal interfaces in pseudospectral 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 pseudospectral method does not provide spectral accuracy in inhomogeneous media. The numerical errors for the upper half of the frequency/wavenumber spectra of the propagating fields are not related to the implementation of the derivative operators but to aliasing effects coming from the multiplication of static material-parameter fields with the dynamic propagating fields. The pseudospectral method can only provide half-spectral accuracy. The same type of spatial aliasing errors is also present for finite-difference schemes. High-order finite differences can provide the same accuracy as the pseudospectral method if the staggered finite-difference derivative operators have a negligible error at four grid points per shortest wavelength and above. Smoothing of the material-parameter field leads to additional reduction in the error-free bandwidth of the propagating fields. Assuming that there is a maximum wavenumber up to which the spectrum of the smoothed model coincides with the implementation using a properly band-limited Heaviside step function, then there exists a local critical wavenumber for the propagating field equal to one 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 which the wavenumber spectrum deviates from an implementation with a band-limited Heaviside step function. The same one-half rule is also applicable in this case.

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