  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.
