Some Novel Aspects of Spectral Methods

Abstract

The papers presented in this section of the book show how extensively spectral methods have changed since the development over twenty years ago of transform methods enabled their efficient computational implementation. Both Fourier and polynomial spectral methods are used routinely today in a wide range of applications including weather simulation, wave dynamics, electromagnetics, and turbulence simulation. In particular, spectral methods have been used almost exclusively over the last decade in simulating turbulent incompressible flows [6]. There are other areas however where the potential of spectral methods has not been yet been fully explored, primarily due to concerns that their performance, both in computational efficiency and convergence rate, may be degraded. This lowering of convergence rate may come from irregularities in the computational domain, presence of discontinuities, inappropriate boundary conditions, etc. We now believe that the integration of good ideas from a variety of numerically-base disciplines may enable the application of spectral techniques to problem classes thought previously to be inappropriate for them. While standard global spectral methods may indeed be inefficient in applications such as compressible turbulence, plasma dynamics, nonlinear optics etc., simple established ideas from other discretization techniques can be employed to make spectral methods also a useful simulation tool in these applications.