BGK-Based Schemes for the Simulation of Compressible Flow

Abstract

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 BGK-scheme 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.