A Kinetic Scheme based on Positivity Preservation with Exact Shock Capture

In this paper, we describe a kinetic model with flexible velocities which are set to satisfy positivity preservation conditions for the Euler equations. Our kinetic model in 1D comprises of two velocities, and both the asymmetrical and the symmetrical models are utilized. Switching between the two models is done based on our own version of kinetic relative entropy and an additional criterion, to ensure an accurate, entropic and robust scheme. In 2D, we describe a novel three-velocity kinetic model, defined to ensure a locally 1D formulation for the resulting macroscopic normal flux. For first order accuracy, we also obtain a limit on the time step which ensures positivity preservation. The resulting numerical scheme captures grid-aligned steady shocks exactly. Several benchmark compressible flow test cases are solved in 1D and 2D to demonstrate the efficacy of the proposed solver.