Kinetic Numerical Methods for the Semiclassical Boltzmann Equation

Euler equations and Navier-Stokes equations can be obtained by taking moments to zero and first order approximations of Chapman-Enskog expansions to classical Boltzmann equation, respectively. The same idea can be implemented to semiclassical Boltzmann equation and one obtains the semiclassical hydrodynamic equations in the limit of Euler and Navier-Stokes orders. The flows of quantum gases described by semiclassical Boltzmann equation includehe effects of particles degeneracy which are not considered in classical Boltzmann equation. Kinetic numerical methods for solving equilibrium and non-equilibrium flows in hydrodynamic limits of semiclassical Boltzmann equation are developed in this study. The BGK relaxation model is used to simplify the scattering term of the semiclassical Boltzmann and the non-equilibrium distribution used in this study was obtained through the first order approximation of Chapman-Enskog expansion to BGK relaxation model. The physical relaxation time in the relaxation model is also derived. First, the quantuminetic beam scheme is developed for solving the flow in equilibrium limit. Formulations to both one and two dimensionality are tested in different cases. In non-equilibrium flow, the quantum gas-kinetic BGK is alsoerived and tested. The numerical validations of the schemes are tested with shock tube. The flow in highly degenerate limit and classical limit condition are simulated. It is shown that in classical thermodynamic conditions the result of classical gas-kinetic BGK scheme can be recovered from the quantum gas-kinetic BGK scheme. Numerical techniques such as WENO, TVD variable extrapolation, and generalized coordinates are adopted to improve and generalized current kinetic schemes.