The PIMS Postdoctoral Fellow Seminar: Samuel Van Fleet

In this talk, we examine a deterministic particle method for the aggregation- diffusion equation and the Landau equation. Particle method solutions are a linear combination of Dirac delta-functions located at certain points. The weights and locations of the Dirac delta-functions evolve with time according to a system of ODEs obtained by a weak formulation of the problem. With- out some sort of regularization, the aggregation-diffusion equation and Landau equation cannot be approximated with a particle method solution. We present a regularization based on a variational formulation of these equations, along with a particle method to approximate solutions to the regularized equations. These particle methods, are structure preserving at the semi-discrete level. There are several choices of numerical method to approximate the solution to the resulting system of ODEs. We show that when a discrete gradient method is used, the resulting particle method is structure preserving at the fully discrete level.