# A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

## TL;DR

This paper introduces a particle-based numerical scheme for linear kinetic equations in the diffusion limit, reformulating the micro-macro system to reduce noise and computational cost, while maintaining asymptotic-preserving properties.

## Contribution

The paper develops a non-stiff reformulation of the micro-macro system enabling standard particle approximations, improving efficiency and noise reduction in kinetic equation simulations.

## Key findings

- Scheme maintains asymptotic-preserving property.
- Significantly reduces noise compared to traditional methods.
- Computational cost decreases as the system approaches the diffusion limit.

## Abstract

In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step $\Delta t$ in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [5] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit.