# A meshfree method for the BGK model for rarefied gas dynamics

## TL;DR

This paper introduces a meshfree semi-Lagrangian method using MLS interpolation to solve the BGK model for rarefied gas dynamics, demonstrating improved accuracy over traditional spline methods through validation against DSMC and Euler solutions.

## Contribution

It presents a novel meshfree semi-Lagrangian approach with MLS interpolation for BGK equations, enhancing solution accuracy in rarefied gas dynamics simulations.

## Key findings

- Meshfree MLS interpolation outperforms spline interpolation.
- The method accurately captures shock tube problems across various mean free paths.
- Results align well with DSMC and Euler solutions for validation.

## Abstract

In this paper we have applied a Semi-Lagrangian schemes with meshfree interpolation, based on a Moving Least Squares (MLS) method, to solve the BGK model for rarefied gas dynamics. Sod's shock tube problems are presented for a large range of mean free paths in one dimensional physical space and three dimensional velocity space. In order to validate the solutions obtained from the meshfree method, we have used the piecewise linear spline interpolation. Furthermore, we have compared the solutions of the BGK model with the solutions obtained from Direct Simulation Monte Carlo (DSMC) method. In the case of a very small mean free path the numerical solutions are compared with the exact solutions of the compressible Euler equations. Overall we found that the meshfree interpolation gives better approximation than the piecewise linear spline interpolation.