An asymptotic preserving method for transport equations with oscillatory scattering coefficients

TL;DR

This paper introduces an asymptotic preserving numerical scheme for transport equations with oscillatory scattering coefficients, effectively capturing both diffusion and homogenization limits through multiscale finite element methods.

Contribution

The paper presents a novel multiscale finite element-based scheme that preserves asymptotic limits in transport equations with oscillatory coefficients, combining diffusion and homogenization modeling.

Findings

The scheme accurately captures the diffusion limit as Knudsen number approaches zero.

It effectively models the homogenization limit as the scattering coefficient's length scale diminishes.

Numerical validations confirm the scheme's robustness across regimes.

Abstract

We design a numerical scheme for transport equations with oscillatory periodic scattering coefficients. The scheme is asymptotic preserving in the diffusion limit as Knudsen number goes to zero. It also captures the homogenization limit as the length scale of the scattering coefficient goes to zero. The proposed method is based on the construction of multiscale finite element basis and a Galerkin projection based on the even-odd decomposition. The method is analyzed in the asymptotic regime, as well as validated numerically.