Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T13:03:15.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Validation for High Order Hyperbolic Moment System of Wigner Equation

Published online by Cambridge University Press:  03 June 2015

Ruo Li ,
Tiao Lu ,
Yanli Wang  and
Wenqi Yao
Ruo Li*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China HEDPS and LMAM, Peking University, Beijing 100871, P.R. China CAPT, Peking University, Beijing 100871, P.R. China
Tiao Lu*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China HEDPS and LMAM, Peking University, Beijing 100871, P.R. China CAPT, Peking University, Beijing 100871, P.R. China
Yanli Wang*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China BICMR, Peking University, Beijing 100871, P.R. China CAPT, Peking University, Beijing 100871, P.R. China
Wenqi Yao*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
*
Email:rli@math.pku.edu.cn
Corresponding author.Email:tlu@math.pku.edu.cn
Email:wangyanliwyl@gmail.com
Email:albee0926@sina.com
Get access

Abstract

A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in [8], to validate the moment system of the Wigner equation. The method developed can keep both mass and momentum conserved, and the variation of the total energy under control though it is not strictly conservative. We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion, and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method. The numerical results indicate that the high order moment system in [6] is a valid model for the Wigner equation, and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.

Keywords

82C1081-0847A57Wigner equationNRxx methodmoment method
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014 , pp. 569 - 595
Copyright
Copyright © Global Science Press Limited 2014

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]Arnold, A. and Ringhofer, C.An operator splitting method for the Wigner-Poisson problem. SIAM Journal on Numerical Analysis, 33(4):pp. 1622–1643, 1996.Google Scholar
[2]Au, J.D., Struchtrup, H., and Torrilhon, M.ETXX — an equation generator for extended thermodynamics. Source available on request via M.Torrilhon@vt.tu-berlin.de.Google Scholar
[3]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, 1954.Google Scholar
[4]Cai, Z., Fan, Y., and Li, R.Globally hyperbolic regularization of Grad’s moment system. To appear in Comm. Pure Appl. Math., 2012.Google Scholar
[5]Cai, Z., Fan, Y., and Li, R.Globally hyperbolic regularization of Grad’s moment system in one dimensional space. Comm. Math Sci., 11(2):547–571, 2013.Google Scholar
[6]Cai, Z., Fan, Y., Li, R., Lu, T., and Wang, Y.Quantum hydrodynamics models by moment closure of Wigner equation. J. Math. Phys., 53:103503, 2012.Google Scholar
[7]Cai, Z. and Li, R.Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation. SIAM J. Sci. Comput., 32(5):2875–2907, 2010.Google Scholar
[8]Cai, Z., Li, R., and Qiao, Z.Globally hyperbolic regularized moment method with applications to microflow simulation. Computer and Fluids, 81:95–109, 2013.Google Scholar
[9]Cai, Z., Li, R., and Wang, Y.An efficient NRxx method for Boltzmann-BGK equation. J. Sci. Comput., 50(1):103–119, 2012.Google Scholar
[10]Cai, Z., Li, R., and Wang, Y.Numerical regularized moment method for high Mach number flow. Commun. Comput. Phys., 11(5):1415–1438,2012.Google Scholar
[11]Cai, Z., Li, R., and Wang, Y.Solving Vlasov equation using NRxx method. To appear in SIAM J. Sci. Comput., 2012.Google Scholar
[12]Degond, P., Méhats, F., and Ringofer, C.Quantum energy-transport and drift-diffusion models. J. Stat. Phys., 118:625–667, 2005.Google Scholar
[13]Degond, P. and Ringhofer, C.Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys., 112:587–628, 2003.Google Scholar
[14]Ferry, D.K. and Goodnick, S. M.Transport in Nanostructures. Cambridge Univ. Press, Cambridge, U.K, 1997.Google Scholar
[15]Frensley, W.R.Wigner function model of a resonant-tunneling semiconductor device. Phys. Rev. B, 36:1570–1580, 1987.Google Scholar
[16]Gardner, C.L.The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math., 54:409–427, 1994.Google Scholar
[17]Goudon, T. and Lohrengel, S.On a discrete model for quantum transport in semi-conductor devices. Transp. Theory Stat. Phys., 31(4-6):471–490, 2002.Google Scholar
[18]Goudon, T.Analysis of a semidiscrete version of the Wigner equation. SIAM J. Numerical Analysis, 40(6):2007–2025, 2002.Google Scholar
[19]Grad, H.On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4):331–407, 1949.Google Scholar
[20]Grad, H.The profile of a steady plane shock wave. Comm. Pure Appl. Math., 5(3):257–300, 1952.Google Scholar
[21]Hillery, M., OConnell, R.F., Scully, M.O., and Wigner, E.P.Distribution functions in physics: Fundamentals. Physics Reports, 106(3):121–167,1984.Google Scholar
[22]Hu, X., Tang, S., and Leroux, M.Stationary and transient simulations for a one-dimensional resonant tunneling diode. Commun. Comput. Phys., 4(5):1034–1050, 2008.Google Scholar
[23]Jensen, K. L. and Buot, F.A.Numerical simulation of intrinsic bistability and high-frequency current oscillations in resonant tunneling structures. Phys. Rev. Lett., 66:1078–1081, Feb 1991.Google Scholar
[24]Jensen, K.L. and Buot, F.A.Numerical aspects on the simulation of IV characteristics and switching times of resonant tunneling diodes. J. Appl. Phys., 67:2153–255, 1990.Google Scholar
[25]Jin, S. and Slemrod, M.Regularization of the Burnett equations via relaxation. J. Stat. Phys, 103(5–6):1009–1033, 2001.Google Scholar
[26]Jüngel, A.A note on current-voltage characteristics from the quantum hydrodynamic equations for semiconductors. Appl. Math. Lett., 10(4):29–34, 1997.Google Scholar
[27]Ansgar Jngel and Shaoqiang Tang. Numerical approximation of the viscous quantum hy-drodynamic model for semiconductors. Appl. Numer. Math., 56(7):899–915, 2006.Google Scholar
[28]Kim, K.-Y. and Lee, Byoungho. On the high order numerical calculation schemes for the wigner transport equation. Solid-State Electronics, 43(12):2243–2245, 1999.Google Scholar
[29]Levermore, C. D.Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5–6):1021–1065, 1996.Google Scholar
[30]Lu, T., Du, G., Liu, X., and Zhang, P.A finite volume method for the multi subband Boltzmann equation with realistic 2D scattering in DG MOSFETs. Commun. Comput. Phys., 10:305–338, 2011.Google Scholar
[31]Dal Maso, G., Le, P. G.Floch, and Murat, F.Definition and weak stability of nonconservative products. J. Math. Pures Appl., 74(6):483–548, 1995.Google Scholar
[32]Nedialkov, M., Kosina, H., Selberherr, S., Ringhofer, C., and Ferry, D. K.Unified particle approach to wigner-boltzmann transport in small semiconductor devices. Physical Review B, page 115319, 2004.Google Scholar
[33]Rhebergen, S., Bokhove, O., and Vegt, J. J. W. van der. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys., 227(3):1887–1922, 2008.Google Scholar
[34]Ringhofer, C.A spectral method for the numerical solution of quantum tunneling phenomena. SIAM J. Num. Anal., 27:32–50, 1990.Google Scholar
[35]Shao, S., Lu, T., and Cai, W.Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport. Commun. Comput. Phys., 9:711–739, 2011.CrossRefGoogle Scholar
[36]Strang, G.On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5(3):506–517, 1968.Google Scholar
[37]Struchtrup, H.Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys. Fluids, 16(11):3921–3934, 2004.Google Scholar
[38]Struchtrup, H. and Torrilhon, M.Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids, 15(9):2668–2680, 2003.Google Scholar
[39]Torrilhon, M.Regularized 13-moment-equations. In Ivanov, M. S. and Rebrov, A. K., editors, Rarefied Gas Dynamics: 25th International Symposium, 2006.Google Scholar
[40]Torrilhon, M., Au, J. D., and Struchtrup, H.Explicit fluxes and productions for large systems of the moment method based on extended thermodynamics. Cont. Mech. and Ther., 15(1):97–111, 2002.Google Scholar
[41]Wigner, E.On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40(5):749–759, Jun 1932.Google Scholar
[42]Zhao, P.J., Woolard, D.L., and Cui, H.L.Multisubband theory for the origination of intrinsic oscillations within double-barrier quantum well systems. Phys. Rev. B, 67:085312, Feb 2003.Google Scholar
[43]Zhou, J.-R. and Ferry, D.K.Simulation of ultra-small GaAs MESFET using quantum moment equations. IEEE Trans. Electron Devices, 39(3):473–478, 1992.Google Scholar