Kinetic numerical methods for solving the semiclassical Boltzmann-BGK equation

The developments of several kinetic numerical methods for solving the semiclassical Boltzmann-BGK equation are presented. The methods considered include the direct solver in phase space and lattice Boltzmann method. For the direct phase space solver, the discrete ordinate methods in velocity space and explicit and implicit high resolution schemes in physical space are combined to yield the desired scheme. A brief overview of the core computational methods which has been originated and evolved over the past 30 years starting at NASA-Ames/Stanford is reviewed. For the semiclassical lattice Boltzmann method, a multiple relaxation time approach is developed. The method is directly derived by projecting the kinetic governing equation onto the tensor Hermite polynomials and various hydrodynamic approximation orders can be achieved. The semiclassical incompressible Navier–Stokes equations can be recovered via a Chapman–Enskog multi-scale expansion. Applications of these kinetic numerical methods to semiclassical hydrodynamic transport involving various kinds of carrier particles are then illustrated by specific examples. The general hydrodynamic transports in gases of arbitrary statistics and in wide flow regimes are emphasized. The results indicate distinct characteristics of the effects of quantum statistics.