Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T20:36:07.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime

Published online by Cambridge University Press:  28 May 2015

Jia Deng
Jia Deng*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR. China
*
*Corresponding author.Email address:[email protected]
Get access

Abstract

As is known, the numerical stiffness arising from the small mean free path is one of the main difficulties in the kinetic equations. In this paper, we derive both the split and the unsplit schemes for the linear semiconductor Boltzmann equation with a diffusive scaling. In the two schemes, the anisotropic collision operator is realized by the “BGK”-penalty method, which is proposed by Filbet and Jin [F. Filbet and S. Jin, J. Comp. Phys. 229(20), 7625-7648, 2010] for the kinetic equations and the related problems having stiff sources. According to the numerical results, both of the schemes are shown to be uniformly convergent and asymptotic-preserving. Besides, numerical evidences suggest that the unsplit scheme has a better numerical stability than the split scheme.

Keywords

35Q2065M12linear semiconductor Boltzmann equationdrift-diffusion limitdiffusive relaxation system“BGK”-penalty method
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 2 , May 2012 , pp. 278 - 296
Copyright
Copyright © Global Science Press Limited 2012

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]Abdallah, N. B., Degond, P. and Genieys, S., An energy-transport model for semiconductors derived from the boltzmann equation, J. Statist. Phys., vol. 84, no. 1-2 (1996), pp. 205–231.Google Scholar
[2]Abdallah, N. B., and Degond, P., On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., vol. 37, no. 7 (1996), pp. 3306–3333.CrossRefGoogle Scholar
[3]Bal, G. and Maday, Y., Coupling of transport and diffusion models in linear transport theory, Math. Model. Numer. Anal., vol. 36, no. 1 (2002), pp. 69–86.CrossRefGoogle Scholar
[4]Cercignani, C., The Boltzmann Equation and Its Appliccations, Springer, Berlin, Wien, 1998.Google Scholar
[5]Degond, P. and Jin, S., A smooth transition model between kinetic and diffusion equations, SIAM. J. Numer. Anal., vol. 42, no. 6 (2005), pp. 2671–2687 (electronic).CrossRefGoogle Scholar
[6]Filbet, F. and Jin, S., A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., vol. 229, no. 20 (2010), pp. 7625–7648.CrossRefGoogle Scholar
[7]Golse, F., Jin, S. and Levermore, C. D., A Domain Decomposition Analysis for a Two-Scale Linear Transport Problem, Math. Model. Numer. Anal., vol. 37, no. 6 (2003), pp. 869–892.CrossRefGoogle Scholar
[8]Hu, J. and Jin, S., On kinetic flux vector splitting schemes for quantum euler equations, kinetic and related models, Kinetic and Related Models., vol. 4, no. 2 (2011), pp. 517–530.CrossRefGoogle Scholar
[9]Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., vol. 21, no. 2 (1999), pp. 441–454.CrossRefGoogle Scholar
[10]Jin, S. and Pareschi, L., Discretization of the multiscale semiconductor boltzmann equation by diffusive relaxation schemes, J. Comput. Phys., vol. 161, no. 1 (2000), pp. 312–330.CrossRefGoogle Scholar
[11]Jin, S., Pareschi, L. and Toscani, G., Diffusive relaxation schemes for multiscale discrete velocity kinetic equations, SIAM J. Numer. Anal., vol. 35, no. 6 (1998), pp. 2405–2439 (electronic).CrossRefGoogle Scholar
[12]Jin, S., Pareschi, L. and Toscani, G., Asymptotic-preserving (AP) schemes for multiscale kinetic equations: a unified approach, Hyperbolic problems: theory, numerics, applications, Vol. I, II (Magdeburg, 2000), 573lC582, Internat. Ser. Numer. Math., 140, 141, Birkhäuser, Basel, 2001.Google Scholar
[13]Jin, S. and Yan, B., A class of asymptotic-preserving schemes for the fokker-planck-landau equation, J. Comput. Phys., vol. 230, no. 17 (2011), pp. 6420–6437.CrossRefGoogle Scholar
[14]Klar, A., Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, SIAM J. Sci. Comput., vol. 19, no. 6 (1998), pp. 2032–2050 (electronic).CrossRefGoogle Scholar
[15]Levermore, C. D., Moment closure hierarchies for kinetic theories, J. Statist. Phys., vol. 83, no. 5-6 (1996), pp. 1021–1065.Google Scholar
[16]Leveque, R., Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1992.CrossRefGoogle Scholar
[17]Markowich, P. A., Ringhofer, C. A. and Schmeiser, C., Semiconductor Equations, Springer-Verlag Wien, New York, 1990.CrossRefGoogle Scholar
[18]Schmeiser, C. and Zwrichmayr, A., Convergence of moment methods for linear kinetic equations, SIAM J. Numer. Anal., vol. 36, no. 1 (1999), pp. 74–88.Google Scholar
[19]Ringhofer, C., Schmeiser, C. and Zwrichmayr, A., Moment methods for the semiconductor Boltzmann equation on bounded position domains, SIAM J. Numer. Anal., vol. 39, no. 3 (2002), pp. 1078–1095.Google Scholar
[20]Tidriri, M., New models for the solution of intermediate regimes in transport theory and radiative transfer: Existence theory, positivity, asymptotic analysis and approximations, J. Statist. Phys., vol. 104, no. 1-2 (2001), pp. 291–325.Google Scholar