Summary.
In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of [21,23]. We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved.
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Received July 10, 2001 / Revised version received October 12, 2001 / Published online January 30, 2002
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Pareschi, L., Toscani, G. & Villani, C. Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93, 527–548 (2003). https://doi.org/10.1007/s002110100384
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DOI: https://doi.org/10.1007/s002110100384