Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Numerical resolution of two-stream kinetic models in a strong aggregative setting is considered. To illustrate our approach, we consider a one-dimensional kinetic model for chemotaxis in hydrodynamic scaling and the high field limit of the Vlasov-Poisson-Fokker-Planck system. A difficulty is that, in this aggregative setting, weak solutions of the limiting model blow up in finite time, and therefore the scheme should be able to handle Dirac measures. It is overcome thanks to a careful discretization of the macroscopic velocity resulting of Vol'pert calculus: accordingly, a new well-balanced and asymptotic preserving numerical scheme is provided. Numerical simulations confirm a good behavior of solutions.