Special Session 100: 

Time diminishing asymptotic preserving scheme for kinetic equations in the diffusion limit

Nicolas Crouseilles
Inria
France
Co-Author(s):    Crestetto, Dimarco, Lemou
Abstract:
In this work, we introduce a new class of numerical schemes for collisional kinetic equations in the diffusion limit. The idea consists in reformulating the problem using a micro-macro decomposition and successively in solving the microscopic part by using asymptotically stable Monte Carlo methods. In addition, the particle method which solves the microscopic part is designed in such a way that the global scheme becomes computationally less expensive as the solution approaches the equilibrium state as opposite to standard methods for kinetic equations which computational cost increases with the number of interactions. At the same time, the statistical error due to the particle part of the solution decreases as the system approach the equilibrium state. This causes the method to degenerate to the sole solution of the macroscopic diffusion equation in the asymptotic regime. In a last part, we will show the behaviors of this new approach in comparisons to standard methods for solving the kinetic equation (Monte Carlo or grid based methods) by testing it on different problems in 2d or 3d.