Abstract
Spectral type methods for the discretization of partial differential equations rely on the approximation of the solution by polynomials of high degree. These methods are proven, both theoretically and numerically, to be of infinite order of accuracy. This infinite order is achieved if the solution is very regular. On the other hand, the Gibbs phenomenon prevents – a priori – the good convergence if the solution is discontinuous. Nevertheless, for systems of conservation laws, the spectral vanishing viscosity method leads to numerical solutions that are “spectrally” close to the projection of the exact solution on the set of polynomials. The idea is then to postprocess the numerical solution in order to extract pertinent physical information. The aim of this paper is to propose and analyse such a postprocessing method based on rational approximants that allows to circumvent the Gibbs phenomenon and can be used as an acceleration device for spectral numerical solution.
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Emmel, L., Kaber, S. & Maday, Y. Padé–Jacobi Filtering for Spectral Approximations of Discontinuous Solutions. Numerical Algorithms 33, 251–264 (2003). https://doi.org/10.1023/A:1025572207222
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DOI: https://doi.org/10.1023/A:1025572207222