I am trying to simulate an unsteady gas flow under vacuum conditions with large variance of temperature. I read many papers about CFD, and know many schemes, such as artificial compressibility, pseudo-compressibility, and so on. But up to now, I am still not sure which scheme is suitable for solving the continuity, NS and energy equations under vacuum. Most of the articles are for incompressible gas or compressible gas, in which gas is compressed and with high Mach number. I am asked to use MatLab to finish the simulation. Could anybody give me any suggestions and any reference information? Thanks a lot!

For rarefied flows (i.e., very low pressures), the NS equations are not valid, as they are based on continuum flow assumptions. In this case, you should probably be looking at a particle-based method, such as DSMC (Direct Simulation Monte Carlo), which is commonly used in such cases. I do not have any references on hand (my work is more in compressible fluid dynamics - Euler/NS), but maybe someone else can help here...