Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T16:38:19.621Z Has data issue: false hasContentIssue false
  • English
  • Français

A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics

Published online by Cambridge University Press:  03 June 2015

Jaw-Yen Yang ,
Bagus Putra Muljadi ,
Zhi-Hui Li  and
Han-Xin Zhang
Jaw-Yen Yang*
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan
Bagus Putra Muljadi*
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan FCS STAE, Université Paul Sabatier, Institut de mathématiques de Toulouse, Toulouse 31400, France
Zhi-Hui Li*
Affiliation:
National Laboratory for Computational Fluid Dynamics, Beijing 100191, China China Aerodynamics Research and Development Center, Mianyang, 621000, China
Han-Xin Zhang*
Affiliation:
National Laboratory for Computational Fluid Dynamics, Beijing 100191, China China Aerodynamics Research and Development Center, Mianyang, 621000, China
*
Corresponding author.Email:yangjy@iam.ntu.edu.tw
Email:muljadi.bp@gmail.com
Email:zhli0097@x263.net
Email:hanxinzhang@tom.com
Get access

Abstract

An accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time. The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved. Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method. The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

Keywords

82-0882B3082B4035Q20Semiclassical Boltzmann-BGK equationweighted essentially non-oscillatorydiscrete ordinate method
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 242 - 264
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abramowitz, M. and Stegun, I. A.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, ninth dover printing, tenth gpo printing edition, 1964.Google Scholar
[2]Ancona, M. G. and Iafrate, G. J.Quantum correction to the equation of state of an electron gas in a semiconductor. Phys. Rev. B, 39(13):9536–9540,May 1989.Google Scholar
[3]Ancona, M. G. and Tiersten, H. F.Macroscopic physics of the silicon inversion layer. Phys. Rev. B, 35(15):7959–7965, May 1987.Google Scholar
[4]Bhatnagar, P. L., Gross, E. P., and Krook, M.A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(3):511–525, May 1954.Google Scholar
[5]Carrillo, J. A., Gamba, I. M., Majorana, A., and Shu, C.-W.A weno-solver for the 1d non-stationary boltzmann Vpoisson system for semiconductor devices. Journal of Computational Electronics, 1:365–370, 2002. 10.1023/A:1020751624960.Google Scholar
[6]Carrillo, J. A., Gamba, I. M., Majorana, A., and Shu, C.-W.A direct solver for 2d non-stationary boltzmann-poisson systems for semiconductor devices: A mesfet simulation by weno-boltzmann schemes. Journal of Computational Electronics, 2:375–380, 2003. 10.1023/B:JCEL.0000011455.74817.35.CrossRefGoogle Scholar
[7]Carrillo, J. A., Gamba, I. M., Majorana, A., and Shu, C.-W.A weno-solver for the transients of boltzmann-poisson system for semiconductor devices: performance and comparisons with monte carlo methods. Journal of Computational Physics, 184(2):498 – 525, 2003.Google Scholar
[8]Chapman, S. and Cowling, T. G.The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases. Cambridge University Press, 1970.Google Scholar
[9]Chen, G.Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons (Mit-Pappalardo Series in Mechanical Engineering). Oxford University Press, USA, Mar. 2005.CrossRefGoogle Scholar
[10]Deng, X. G. and Zhang, H. X.Developing high-order weighted compact nonlinear schemes. Journal of Computational Physics, 165:24–44, 2000.Google Scholar
[11]Fatemi, E. and Odeh, F.Upwind finite difference solution of boltzmann equation applied to electron transport in semiconductor devices. Journal of Computational Physics, 108(2):209 – 217, 1993.CrossRefGoogle Scholar
[12]Gardner, C. L.The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math., 54(2):409–427, 1994.CrossRefGoogle Scholar
[13]Harten, A.High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49(3):357 – 393, 1983.Google Scholar
[14]Harten, A. and Lax, P. D.On a class of high resolution total-variation-stable finite-difference schemes. SIAM Journal on Numerical Analysis, 21(1):pp. 1–23, 1984.Google Scholar
[15]Hsieh, T.-Y. and Yang, J.-Y.Thermal conductivity modeling of circular-wire nanocomposites. Journal of Applied Physics, 108:044306, 2010.Google Scholar
[16]Huang, A. B. and Giddens, D. P.The Discrete Ordinate Method for the Linearized Boundary Value Problems in Kinetic Theory of Gases. In Brundin, C. L., editor, Rarefied Gas Dynamics, Volume 1, pages 481–+, 1967.Google Scholar
[17]Jiang, G.-S., Levy, D., Lin, C.-T., Osher, S., and Tadmor, E.High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 35(6):pp. 2147–2168, 1998.CrossRefGoogle Scholar
[18]Jin, S.Runge-kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys, 122:51–67, 1995.Google Scholar
[19]Jin, S. and Levermore, C. D.Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys, 126:449–467, 1996.Google Scholar
[20]Jin, S. and Xin, Z. P.The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Communications on Pure and Applied Mathematics, 48:235, 2006.Google Scholar
[21]Kadanoff, L. P. and Baym, G.Quantum Statistical Mechanics. Benjamin, New York, 1962.Google Scholar
[22]Lax, P. D. and Liu, X. D.Solution of two dimensional riemann problem of gas dynamics by positive schemes. SIAM J. Sci. Comput, 19:319–340, 1995.Google Scholar
[23]Li, Z.-H. and Zhang, H.-X.Numerical investigation from rarefied flow to continuum by solving the boltzmann model equation. Intern. J. Numer. Fluids, 42:361–382, 2003.Google Scholar
[24]Li, Z.-H. and Zhang, H.-X.Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. Journal of Computational Physics, 193:708–738, 2004.Google Scholar
[25]Lundstrom, M.Fundamentals of Carrier Transport. Cambridge University Press, 2nd edition, 2000.Google Scholar
[26]Majorana, A. and Pidatella, R. M.A finite difference scheme solving the boltzmann-poisson system for semiconductor devices: Volume 174, number 2 (2001), pages 649–668. Journal of Computational Physics, 177(2):450 – 450, 2002.Google Scholar
[27]Markowich, P. A., Ringhofer, C. A., and Schmeiser, C.Semiconductor Equations. Springer, 1 edition, 2002.Google Scholar
[28]Nikuni, T. and Griffin, A.Hydrodynamic damping in trapped bose gases. Journal of Low Temperature Physics, 111:793–814, 1998.Google Scholar
[29]Pattamatta, A. and Madnia, C. K.Modeling electron-phonon nonequilibrium in gold films using boltzmann transport model. Journal of Heat Transfer, 131:082401–1, 2009.CrossRefGoogle Scholar
[30]Scaldaferri, S., Curatola, G., and Iannaccone, G.Direct solution of the boltzmann transport equation and poisson schrodinger equation for nanoscale mosfets. IEEE Transaction on Electron Devices, 54:2901, 2007.Google Scholar
[31]Schultz-Rinne, C. W., Collins, J. P., and Glaz, H. M.Numerical solution of the riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput., 14(6):1394–1414, 1993.CrossRefGoogle Scholar
[32]Shi, Y. H. and Yang, J. Y.A gas kinetic bgk scheme for semiclassical boltzmann hydrodynamic transport. Journal of Computational Physics, 227(22):9389 – 9407, 2008.Google Scholar
[33]Shizgal, B.A gaussian quadrature procedure for use in the solution of the boltzmann equation and related problems. Journal of Computational Physics, 41(2):309 – 328, 1981.Google Scholar
[34]Uehling, E. A. and Uhlenbeck, G. E.Transport phenomena in einstein-bose and fermi-dirac gases. i. Phys. Rev., 43(7):552–561, Apr 1933.Google Scholar
[35]Leer, B. van. Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov’s method. Journal of Computational Physics, 32(1):101 – 136, 1979.Google Scholar
[36]Wigner, E.On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40(5):749–759, Jun 1932.Google Scholar
[37]Woolard, D. L., Tian, H., Littlejohn, M. A., Kim, K. W., Trew, R. J., Ieong, M. K., and Tang, T. W.Construction of higher-moment terms in the hydrodynamic electron transport model. Journal of Applied Physics, 74(10):6197 –6207, nov 1993.Google Scholar
[38]Xu, Z. and Shu, C.-W.Anti-diffusive flux corrections for high order finite difference weno schemes. Journal of Computational Physics, 205(2):458 – 485, 2005.Google Scholar
[9]Yang, J. Y., Hsieh, T. Y., and Shi, Y. H.Kinetic flux vector splitting schemes for ideal quantum gas dynamics. SIAM J. Sci. Comput., 29(1):221–244,2007.Google Scholar
[40]Yang, J. Y. and Huang, J. C.Rarefied flow computations using nonlinear model boltzmann equations. Journal of Computational Physics, 120(2):323 – 339, 1995.CrossRefGoogle Scholar
[41]Yang, J. Y. and Shi, Y. H.A kinetic beam scheme for ideal quantum gas dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 462(2069):1553–1572, 2006.Google Scholar
[42]Zhang, H.-X.Non-oscillatory and non-free-parameter dissipation difference scheme. Acta Aerodynamica Sinica, 9(6):143–165, 1988.Google Scholar