On the construction of BGK-type schemes for compressible flow simulations

Following van Leer's MUSCL idea, a numerical scheme can be regarded as consisting of two key steps: a reconstruction step followed by a gas evolution step. We present a gas-kinetic method based on the collisional BGK model which provides an alternative to Riemann solvers for the gas evolution step. An advanced BGKscheme is derived under quite general assumptions on the initial conditions. The new formulation uses interpolation of the characteristic variables in the reconstruction step and a BGK-type flow solver in the gas evolution step. The scheme satisfies both an entropy condition and a positivity condition, which guarantees a positive density and temperature at the cell interface during a complete time step. Numerical results for one-dimensional and two-dimensional test cases are presented to show the accuracy and robustness of the proposed approach.