Collective Behavior: Macroscopic versus Kinetic Descriptions

Structure Preserving Methods for Fokker-Planck-type Equations

Hailiang Liu

Iowa State University

Kinetic Fokker-Planck equations arise in many applications, and thus there has been considerable interest in the development of accurate numerical methods to solve them. The peculiar feature of these models is that the transient solution converges to certain equilibrium when time becomes large. For the numerical method to capture the long-time pattern of the underlying solution, some structure preserving methods have been designed to preserve physical properties exactly at the discrete level. I shall explain the main ideas and challenges through several examples, including the Fokker-Planck equation of the dumbbell model for polymers, a reaction-diffusion-advection equation for the evolution of biased dispersal of population dynamics, and a direct competitive selection model. Numerical results are reported to illustrate the capacity of the proposed algorithms.