Commun. Comput. Phys., 15 (2014), pp. 1291-1319.


Implicit-Explicit Runge-Kutta Schemes for the Boltzmann-Poisson System for Semiconductors

Giacomo Dimarco 1, Lorenzo Pareschi 2, Vittorio Rispoli 2*

1 Universite de Toulouse; UPS, INSA, UT1, UTM; CNRS, UMR 5219; Institut de Mathematiques de Toulouse; F-31062 Toulouse, France.
2 Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 35, 44121 Ferrara, Italy.

Received 9 May 2013; Accepted (in revised version) 15 November 2013
Available online 14 February 2014
doi:10.4208/cicp.090513.151113a

Abstract

In this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diffusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diffusion equation. Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diffusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit. We show different numerical results which permit to appreciate the performances of the proposed schemes.

AMS subject classifications: 35B25, 65M06, 76R50, 82C70

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Key words: IMEX-RK methods, asymptotic preserving methods, semiconductor Boltzmann equation, drift-diffusion limit.

*Corresponding author.
Email: giacomo.dimarco@math.univ-toulouse.fr (G. Dimarco), lorenzo.pareschi@unife.it (L. Pareschi), vittorio.rispoli@unife.it (V. Rispoli)
 

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