The Pseudospectral Method and Discrete Spectral Analysis

One of the focal points of the research at the Centre for Nonlinear Studies is related to the numerical simulation of the emergence, propagation and interaction of solitary waves and solitons in nonlinear dispersive media. Based on the discrete Fourier transform the pseudospectral method can be used for the numerical integration of the model equations, and the Fourier transform related discrete spectral characteristics for the analysis of the numerical results. The latter approach is called discrete spectral analysis. The main advantage of the pseudospectral method compared with the finite difference method is related to the computational costs. On the other hand, the Fourier transform related spectral characteristics carry additional information about the internal structure of the waves, which can be used for the analysis of the time-space behaviour of the numerical solutions. In the present paper several practical aspects of the application of the Fourier transform based pseudospectral method for the numerical integration of equations of different types are discussed and several examples of applications of the discrete spectral analysis are introduced.