Young Researchers Workshop:
Kinetic and macroscopic models for complex systems

Numerical computation of the Boltzmann equation with singular intermolecular potentials and multi species, multi-energy level gases

Jeff Haack

The University of Texas at Austin


We present a conservative spectral scheme for Boltzmann collision operators. This formulation is derived from the weak form of the Boltzmann equation, which can represent the collisional term as a weighted convolution in Fourier space. The weights contain all of the information of the collision mechanics and can be precomputed. I will present some results for isotropic (in angle) interations, such as hard spheres and Maxwell molecules. We have recently extended the method to take into account anisotropic scattering mechanisms arising from intermolecular potential interactions between particles, and we use this method to compute the Boltzmann equation with a screened Coulomb potential. In particular, we study the rate of convergence of the Fourier transform for the Boltzmann collision operator in the grazing collisions limit to the Fourier transform for the limiting Landau collision operator. We show that the decay rate to equilibrium depends on the parameters associated with the collision cross section, and specifically study the differences between the classical Rutherford scattering angular cross section, which has logarithmic error, and an artificial one with a linear error. I will also present recent work extending this method for multispecies gases and gas with internal degrees of freedom, which introduces new challenges for conservation and introduces inelastic collisions to the system.