A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport

A class of gas-kinetic BGK schemes for solving quantum hydrodynamic transport based on the semiclassical Boltzmann equation with the relaxation time approximation is presented. The derivation is a generalization to the development of Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, from gas-kinetic theory, J. Comput. Phys. 171 (2001) 289–335] for the classical gas. Both Bose–Einstein and Fermi–Dirac gases are considered. Some new features due to the quantum equilibrium distributions are delineated. The first-order Chapman–Enskog expansion of the quantum BGK-Boltzmann equation is derived. The coefficients of shear viscosity and thermal conductivity of a quantum gas are given. The van Leer’s limiter is used to interpolate and construct the distribution on interface to achieve second-order accuracy. The present quantum gas-kinetic BGK scheme recovers the Xu’s scheme when the classical limit is taken. Several one-dimensional quantum gas flows in a shock tube are computed to illustrate the present method.