SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Spectral Method for the Transport Equation - Fast Expansion into a Suitable Basis

H. Dietert and A. Iserles

Abstract
We wish to solve equations with transport terms and Dirichlet boundary conditions by spectral methods, combined with splitting in time. The condition for stability is that the differentiation matrix is skew-symmetric, but this is unattainable by any polynomial basis. Using a stretched Fourier transformation, I derive an orthogonal basis with a tridiagonal skew-symmetric differentiation matrix together with an algorithm for fast expansion into this basis for analytic functions. This expansion converges geometrically fast and we only need $O(n \log n)$ operations to compute the first $n$ coefficients.

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