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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.

MSC:

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

Citations:

Zbl 1058.76056
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[1] Beyer, J., Effect of a convex boundary on a rarefied classical or quantum gas: flow past a stationary boundary and the effect of an oscillating boundary on the gas, Phys. rev. B, 33, 511-525, (1986)
[2] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases, I. small amplitude processes in charged and neutral one-component system, Phys. rev., 94, 3, 511-525, (1954) · Zbl 0055.23609
[3] Bird, G.A., Molecular gas dynamics and the direct simulation of gas flow, (1994), Oxford University Press Oxford
[4] Chapman, S.; Cowling, T.G., The mathematical theory of non-uniform gases, (1970), Cambridge University Press · Zbl 0098.39702
[5] Chen, G., Nanoscale energy transport and conversion, (2005), Oxford University Press
[6] Chou, S.Y.; Baganoff, D., Kinetic flux-vector splitting for the navier – stokes equations, J. comput. phys., 130, 217-230, (1997) · Zbl 0873.76057
[7] Degond, P.; Ringhofer, C., Quantum moment hydrodynamics and the entropy principle, J. stat. phys., 112, 587-628, (2003) · Zbl 1035.82028
[8] Garcia, A.L.; Wagner, W., Direct simulation Monte Carlo method for the uehling – uhlenbeck – boltzmann equation, Phys. rev. E, 68, 056703, (2003)
[9] Gardner, C.L., The quantum hydrodynamic model for semiconductor devices, SIAM J. appl. math., 54, 409-427, (1994) · Zbl 0815.35111
[10] Gardner, C.L.; Ringhofer, C., The chapman – enskog expansion and the quantum hydrodynamic model for semiconductor devices, VLSI des., 10, 415-435, (2000)
[11] Griffin, A.; Wu, W.C.; Stringari, S., Hydrodynamic modes in a trapped Bose gas above the bose – einstein transition, Phys. rev. lett., 78, 1838, (1997)
[12] Hsieh, T.Y.; Yang, J.Y.; Shi, Y.H., Kinetic flux vector splitting schemes for ideal quantum gas dynamics, SIAM J. sci. comput., 29, 221-244, (2007) · Zbl 1331.82036
[13] Huang, K., Statistical mechanics, (1987), Wiley New York · Zbl 1041.82500
[14] Kadanoff, L.P.; Baym, G., Quantum statistical mechanics, (1962), Benjamin New York, (Chapter 6) · Zbl 0115.22901
[15] Lundstrom, M., Fundamentals of carrier transport, (2000), Cambridge University Press Cambridge
[16] Nikuni, T.; Griffin, A., Hydrodynamic damping in trapped Bose gases, J. low temp. phys., 111, 793-814, (1998)
[17] Ohwada, T., The kinetic scheme for the full-Burnett equations, J. comput. phys., 201, 315-332, (2004) · Zbl 1195.76354
[18] Santos, P.F.; LĂ©onard, J.; Wang, J.; Barrelet, C.J.; Perales, F.; Rasel, E., Bose – einstein condensation of metastable helium, Phys. rev. lett., 86, 3459-3462, (2001)
[19] Uehling, E.A.; Uhlenbeck, G.E., Transport phenomena in einstein – bose and fermi – dirac gases. I, Phys. rev., 43, 552, (1933) · Zbl 0006.33404
[20] Wigner, E., On the quantum correction for thermodynamic equilibrium, Phys. rev., 40, 749-759, (1932) · JFM 58.0948.07
[21] Xu, K.; Prendergast, K.H., Numerical navier – stokes solutions from gas-kinetic theory, J. comput. phys., 114, 9, (1994) · Zbl 0810.76059
[22] K. Xu, Gas-kinetic scheme for unsteady compressible flow simulations, in: VKI for Fluid Dynamics Lecture Series, 1998-2003, 1998.
[23] Xu, K., 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, 289-335, (2001) · Zbl 1058.76056
[24] Xu, K.; Mao, M.; Tang, L., A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow, J. comput. phys., 203, 405-421, (2005) · Zbl 1143.76553
[25] Yang, J.Y.; Huang, J.C., Rarefied flow computations using nonlinear model Boltzmann equations, J. comput. phys., 120, 323-339, (1995) · Zbl 0845.76064
[26] Yang, J.Y.; Shi, Y.H., A kinetic beam scheme for the ideal quantum gas dynamics, Proc. roy. soc. lond. A, 462, 1553, (2006) · Zbl 1149.82341
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