Numerical transport process of splitting kinetic schemes in the Navier-Stokes-Fourier limit

The Boltzmann equation is the fundamental governing equation in rarefied gas dynamics. Due to the complexity of Boltzmann collision term, operator splitting treatment is commonly adopted, where the Boltzmann equation is split into a convection equation for particles' free transport and an ordinary differential equation for particles' collision. However, this split treatment will introduce numerical error proportional to the time step, which may contaminate the physical solution in the near continuum regime. Therefore, for a multiscale kinetic method, the asymptotic preserving property to obtain the Navier-Stokes-Fourier (NSF) solution in the hydrodynamic limit is very important. In this paper, we analyse the effective relaxation time from different evolution processes of several kinetic schemes and investigate their capabilities to recover the NSF solution. The general requirement on a splitting kinetic method for the NSF solution has been presented. Numerical validation has been carried out, which shows good agreement with the theoretical analysis.