Spectral method

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In numerical mathematics , the spectral method is a method for solving partial differential equations , such as the Navier-Stokes equations , using global shape functions . Approach functions can e.g. B. Fourier series or Chebyshev polynomials . In the course of a numerical solution process, the physical representation of a problem is transformed into the spectral range. The unknowns are then no longer physical quantities, such as a discrete speed or temperature curve, but the spectral coefficients of the global approach function. Hence the term spectral method as a superordinate term. The fast Fourier transformation (FFT) is an efficient method for forward and backward transformation . Coefficients of Chebyshev polynomials can also be determined with this, provided the Gauss-Lobatto points are used as support points, since the transformation is then limited to the real parts of a Fourier transformation.

These methods show favorable convergence properties for tasks whose solutions have a high degree of smoothness. In addition, the approach function should be adapted to the physical problem. Fourier series are suitable for periodic boundary conditions. In the case of fixed, non-periodic values ​​at the edges of the solution area, approach functions should be used that can naturally reproduce these curves. If a finer discretization is also required at the edges, Chebyshev polynomials are advantageous here (see Gauss-Lobatto points). If, however, Fourier series are used instead, Gibbs oscillations are to be expected. In addition, the equidistant grid of a Fourier approximation by an FFT does not do justice to the finer resolution at the edges.

A typical case in which both approximations are used is the three-dimensional planar channel flow. Due to the high gradients near the wall and the clearly non-periodic behavior on the wall, Chebyshev polynomials are used in the direction of the wall normal. In the main flow and span directions, however, periodic boundary conditions are required in order to numerically model an infinitely extended flat channel.

The disadvantage is that the spectral method can lead to linear systems of equations with fully populated and asymmetrical matrices . Iterative procedures are then required for the solution . The multi-grid method has proven itself here. However, there are processes in which a clever re-sorting leads to a matrix with a band structure. LU decomposition is advantageous here. Techniques for domain decomposition (Engl. Domain decomposition ) are also of interest.

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  • Channelflow is a program under GPL based on the algorithm from Chapter 7.3 of the book "Spectral Methods in Fluid Dynamics"