A fast Fourier spectral method for the homogeneous Boltzmann equation with non-cutoff collision kernels

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Original languageEnglish
Article number109806
Journal / PublicationJournal of Computational Physics
Online published31 Aug 2020
Publication statusPublished - 15 Dec 2020


We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lot of difficulties in both analysis and numerics, hence are often cut off in most studies (the well-known Grad's angular cutoff assumption). We demonstrate that the general framework of the fast Fourier spectral method developed in [9], [14] can be extended to handle the non-cutoff kernels, achieving the accuracy/efficiency comparable to the cutoff case. We also show through several numerical examples that the solution to the non-cutoff Boltzmann equation enjoys the smoothing effect, a striking property absent in the cutoff case.

Research Area(s)

  • Boltzmann equation, Non-cutoff collision kernel, Singularity, Fractional Laplacian, Fourier spectral method, Fast fourier transform