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Conclusion

The Fourier method can be considered as the limit of the finite-difference method as the length of the operator tends to the number of points along a particular dimension. The space derivatives are calculated in the wavenumber domain by multiplication of the spectrum with $ ik$ . The inverse Fourier transform results in an exact space derivative up to the Nyquist frequency. The use of Fourier transform imposes some constraints on the smoothness of the functions to be differentiated. Discontinuities lead to Gibbs' phenomenon. As the Fourier transform requires periodicity this technique is particular useful where the physical problems are periodical (e.g. angular derivatives in cylindrical problems).

By introducing fractional order of the Laplacian operator, we are able to perform wave simulation in attenuative medium. The computation of fractional Laplacian can be easily carried out by using Fourier pseudo spectral method, which is computational accurate and efficient enough without any additional storage when considering a constant Q model.


next up previous [pdf]

Next: Absorbing boundary condition Up: Fourier pseudo spectral method Previous: Fractional Laplacian

2021-08-31