Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

Convergence rates for the Boltzmann equation for Coulombic interactions to Landau equation: analysis and simulations

Irene Gamba

University of Texas at Austin

The Fourier transform of both, the non-linear Boltzmann collisional integral and the Landau operator, derived from the weak formulation of their respective equations, can be represent as weighted convolutions in Fourier space.

We will present an overview of analytical issues and novel numerical methods consisting on spectral-conservative schemes for these equations that preserve the expected conserved properties of the described phenomena, while enabling rigorous stability, convergence and error analysis.

Within this framework, we study the rate of convergence of the Fourier transforms of the difference of the Boltzmann grazing collision and Landau operators, for a large family of power law Boltzmann angular scattering cross sections that include the Rutherford potential as the critical limit for grazing conditions. We analytically show that the decay rate to equilibrium depends on the parameters associated to these collision cross section.

This is work in collaboration with Ricardo Alonso, Jeff Haack and S. Harsha Tharkabhushanam.