A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport. (English) Zbl 1231.82056

Summary: 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 K. Xu [J. Comput. Phys. 171, No. 1, 289–335 (2001; Zbl 1058.76056)] 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.


82C40 Kinetic theory of gases in time-dependent statistical mechanics
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
76M28 Particle methods and lattice-gas methods


Zbl 1058.76056
Full Text: DOI


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