Moment Methods for the Semiconductor Boltzmann Equation on Bounded Position Domains

Galerkin methods for the semiconductor Boltzmann equation based on moment expansions for a discretization in the velocity direction are studied. For the moment equations, boundary conditions are proposed, which are analogues to inflow and, respectively, reflecting boundary conditions for the Boltzmann equation. Stability and an error estimate are proved for an expansion in terms of Hermite polynomials. Finally, an adaptive numerical implementation is introduced and results of numerical experiments are presented.

[1]  C. D. Levermore,et al.  Moment closure hierarchies for kinetic theories , 1996 .

[2]  C. Bardos,et al.  DIFFUSION APPROXIMATION AND COMPUTATION OF THE CRITICAL SIZE , 1984 .

[3]  Lorenzo Pareschi,et al.  Discretization of the Multiscale Semiconductor Boltzmann Equation by Diffusive Relaxation Schemes , 2000 .

[4]  Axel Klar,et al.  Asymptotic-Induced Domain Decomposition Methods for Kinetic and Drift Diffusion Semiconductor Equations , 1998, SIAM J. Sci. Comput..

[5]  Shi Jin,et al.  Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations , 2000, SIAM J. Numer. Anal..

[6]  Axel Klar,et al.  A Numerical Method for Kinetic Semiconductor Equations in the Drift-Diffusion Limit , 1999, SIAM J. Sci. Comput..

[7]  Hansjörg Dirschmid,et al.  Einführung in die mathematischen Methoden der Theoretischen Physik , 1976 .

[8]  Christian Schmeiser,et al.  Convergence of Moment Methods for Linear Kinetic Equations , 1998 .

[9]  Christian A. Ringhofer,et al.  Space-Time Discretization of Series Expansion Methods for the Boltzmann Transport Equation , 2000, SIAM J. Numer. Anal..

[10]  I. Müller,et al.  Rational Extended Thermodynamics , 1993 .

[11]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[12]  Lorenzo Pareschi,et al.  An Implicit Monte Carlo Method for Rarefied Gas Dynamics , 1999 .

[13]  A. Klar.  An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit , 1998 .

[14]  Alexandre J. Chorin.  Numerical solution of Boltzmann's equation , 1972 .

[15]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .