A conservative discrete ordinate method for solving semiclassical Boltzmann-BGK equation with Maxwell type wall boundary condition

A robust and efficient numerical method in phase space combining the conservative discrete ordinate method and high order spatial discretization is proposed for the semiclassical Boltzmann equation with Bhatnagar-Gross-Krook (BGK) relaxation time approximation. A general Maxwell type partially diffuse and partially specular reflection solid wall boundary condition for modeling the gas-surface interactions for semiclassical rarefied gas flows is devised. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to evolve the solution in physical space and time. The conservation property of the BGK collision integral is ensured to be fulfilled at the discrete quadrature level approximation. Both specular and diffuse reflection boundary conditions are implemented. Extensive computations of one- and two-dimensional semiclassical rarefied gas dynamical flows covering wide range of flow regimes for three statistics are presented to illustrate the method and boundary conditions treatment. The effect of accommodation coefficient on the rarefied flow field is also examined.