Gas-kinetic schemes for the compressible Euler equations: Positivity-preserving analysis

Numerical schemes based on the collisional BGK model have been developed in recent years. In this paper, we investigate the first-order BGK schemes for the Euler equations. Particular attention is given to finding CFL-like conditions under which the schemes are positivity-preserving (i.e. density and internal energy remain nonnegative). The first-order BGK schemes are linear combinations of collisionless (i.e. kinetic flux-splitting scheme) and collisional approach. We show that the collisionless approach preserves the positivity of density and internal energy under the standard CFL condition. Although the collisionless approach has the positivity-preserving property, it introduces large intrinsic dissipation and heat conductions since the corresponding scheme is based on two half Maxwellians. In order to reduce the viscous error, one obvious method is to use an exact Maxwellian, which leads to the collisional Boltzmann scheme. An CFL-like condition is also found for the collisional approach, which works well for the test problems available in literature. However, by considering a counterexample we find that the collisional approach is not always positivity-preserving. The BGK type schemes are formed by taking the advantages of both approaches, i.e. the less dissipative scheme (collisional) and the more dissipative but positivity-preserving scheme (collisionless).