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

Abstract

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 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.

References in corpus (4)

• Fast algorithms for computing the Boltzmann collision operator
• A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit
• Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff
• Numerical computation for the non-cutoff radially symmetric homogeneous boltzmann equation

Cited by in corpus (4)

• A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation
• On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials
• A fast Fourier spectral method for wave kinetic equation
• Continuous data assimilation for hydrodynamics: consistent discretization and application to moment recovery