Some examples of kinetic scheme whose diffusion limit is Il'in's exponential-fitting

TL;DR

This paper demonstrates that well-balanced numerical schemes for certain kinetic models asymptotically approach Il'in's exponential-fitting discretization under diffusive scaling, ensuring accurate macroscopic approximations.

Contribution

It proves the asymptotic preserving property of well-balanced schemes for kinetic equations, linking them to Il'in's exponential-fitting method through S-matrix decomposition.

Findings

Well-balanced schemes relax to exponential-fitting discretization in diffusive limit.

The S-matrix decomposition is key to establishing the asymptotic preserving property.

The approach applies to models like radiative transfer, chemotaxis, and plasma physics.

Abstract

This paper is concerned with diffusive approximations of peculiar numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov-Fokker-Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a '' scattering S-matrix '' , itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il'in/Scharfetter-Gummel's '' exponential fitting '' discretization. We prove that the well-balanced schemes relax, within a parabolic rescaling, towards the Il'in exponential-fitting discretization by means of an appropriate decomposition of the S-matrix. This is the so-called asymptotic preserving (or uniformly accurate) property.