Spectral Methods

   

• A very large class of computational problems can be handled exquisitely well by Fourier Transform Methods or Spectral Methods as they're often referred to.
• For some problems Fourier Transform is merely a convenient tool to manipulate data.
• For some problems Fourier Transform itself (i.e., the power spectrum) is of special interest.
• For most linear PDEs Spectral Methods are probably the most suitable: they do not suffer inefficiencies and inaccuracies of explicit methods, and they are much easier to program than implicit methods.
• Many extremely difficult non-linear problems can be also solved by a combination of Fourier Transform and various iterative procedures.
  • The Earth Dynamo     problem was solved numerically by Gary A. Glatzmaier from the   Los Alamos National Laboratory, who used a variant of a Spectral Method.
    • ``A Three-Dimensional Convective Dynamo Solution with Rotating and Finitely Conducting Inner Core and Mantle'', G. A. Glatzmaier and P. H. Roberts (UCLA), Physics   of the Earth and Planetary Interiors, 1994.
    • Over 2,000,000 numerical time steps were needed to simulate the 40,000 years evolution of the magnetic field. The program fitted into 300MB memory (very small) and too 2,000 CPU hours on a Cray C90.
    • A dynamic field reversal   has been observed during simulation.