A kinetic theory based scheme for the numerical solution of the BGK-Burnett equations for hypersonic flows in the continuum-transition regime

In this paper a kinetic theory based upwind algorithm for the BGK-Burnett equations is presented. The Boltzmann equation, with Bhatnagar-Gross-Krook (BGK) approximation for the collision integral, describes the spatial and temporal variations of the second-order distribution function which forms the basis of this formulation. The second order distribution function is derived by considering the first three terms in the Chapman-Enskog expansion and using the Navier-Stokes equations to express the material derivatives, present in the second-order terms, in terms of the spatial derivatives. The Burnett equations are derived by taking moments of the BGK-Boltzmann equation with the collision invariant vector. A Kinetic Wave/Particle Split scheme for the BGK-Burnett equations is derived by taking moments of the upwind discretized Boltzmann equation. This algorithm is applied to a 1-D shock tube problem and a hypersonic shock structure problem. This is the first time that a kinetic-theory based method has been developed for solving the Burnett equations.