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

Published online by Cambridge University Press:  03 June 2015

Giacomo Dimarco ,
Lorenzo Pareschi  and
Vittorio Rispoli
Giacomo Dimarco*
Affiliation:
Université de Toulouse; UPS, INSA, UT1, UTM; CNRS, UMR 5219; Institut de Mathématiques de Toulouse; F-31062 Toulouse, France
Lorenzo Pareschi*
Affiliation:
Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 35, 44121 Ferrara, Italy
Vittorio Rispoli*
Affiliation:
Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 35, 44121 Ferrara, Italy
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
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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.

Keywords

35B2565M0676R5082C70IMEX-RK methodsasymptotic preserving methodssemiconductor Boltzmann equationdrift-diffusion limit
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 5 , May 2014 , pp. 1291 - 1319
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Berger, M. J. and Collela, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comp. Phys., 82 (1989), 62–84.CrossRefGoogle Scholar
[2]Anile, A. M., Romano, V. and Russo, G., Hyperbolic hydrodynamical model of carrier transport in semiconductors, VLSI Design, 8 (1998), 521–525.CrossRefGoogle Scholar
[3]Ascher, U., Ruuth, S. and Spitheri, R. J., Implicit-explicit Runge-Kutta methods for time dependent Partial Differential Equations, Appl. Num. Math. 25 (1997), 151–167.CrossRefGoogle Scholar
[4]Bal, G. and Maday, Y., Coupling of transport and diffusion models in linear transport theory, Math. Model. Num. Anal., 36 (2002), 69–86.Google Scholar
[5]Bennoune, M., Lemou, M. and Mieussens, L., Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asympotics, J. Comp. Phys., 227 (2008), 3781–3803.Google Scholar
[6]Boscarino, S., Pareschi, L. and Russo, G., Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, J. Sci. Comp. 35 (2013), 22–51.Google Scholar
[7]Buet, C. and Cordier, S., An asymptotic preserving scheme for hydrodynamics radiative transfer models, Numerische Math., 108 (2007), 199–221.CrossRefGoogle Scholar
[8]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, J. Comp. Phys., 184 (2003), 498–525.CrossRefGoogle Scholar
[9] J. A.Carrillo, Gamba, I.M., Majorana, A. and Shu, C. W., 2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods, J. Comp. Phys., 214 (2006), 55–80.Google Scholar
[10]Chorin, A. J., Numerical solution of Boltzmann’s equation, Comm. Pure Appl. Math., 25 (1972), 171–186.CrossRefGoogle Scholar
[11]Carpenter, M. H. and Kennedy, C. A., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Num. Math., 44 (2003), 139–181.Google Scholar
[12]Crouseilles, N. and Lemou, M., An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kin. Rel. Mod., 4 (2011), 441–477.Google Scholar
[13]Degond, P. and Jin, S., A smooth transition model between kinetic and diffusion equations, J. Num. Anal., 42 (2005), 2671–2687.Google Scholar
[14]Degond, P. and Schmeiser, C., Kinetic boundary layers and fluid-kinetic coupling in semiconductors, Transp. Theory Stat. Phys., 28 (1999), 31–55.Google Scholar
[15]Degond, P., Dimarco, G. and Mieussens, L., A moving interface method for dynamic kinetic-fluid coupling, J. Comp. Phys., 227 (2007), 1176–1208.CrossRefGoogle Scholar
[16]Degond, P., Dimarco, G. and Mieussens, L., A multiscale kinetic-fluid solver with dynamic localization of kinetic effects, J. Comp. Phys., 229 (2010), 4907–4933.CrossRefGoogle Scholar
[17]Dimarco, G. and Pareschi, L., Exponential Runge-Kutta methods for stiff kinetic equations, J. Num. Anal., 49 (2011), 2057–2077.Google Scholar
[18]Dimarco, G. and Pareschi, L., High order asymptotic preserving schemes for the Boltzmann equation, Math, C. R., 350 (2012), 481–486.Google Scholar
[19]Dimarco, G. and Pareschi, L., Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations, J. Num. Anal. 51, (2013), 1064–1087.Google Scholar
[20]Fatemi, E., Jerome, J. W. and Osher, S., Solution of the hydrodynamic device model using high order non oscillatory shock capturing algorithm, Trans. Comp. A. Des., 10 (1991), 232–244.Google Scholar
[21]Filbet, F. and Jin, S., A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625–7648.CrossRefGoogle Scholar
[22]Gosse, L. and Toscani, G., An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math., Acad. Sci. Paris, 334 (2002), 337–342.Google Scholar
[23]Gosse, L. and Toscani, G., Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation, Num. Math., 98 (2004), 223–250.CrossRefGoogle Scholar
[24]Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, New York, 1987.Google Scholar
[25]Jin, S., Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations, J. Sci. Comp., 21 (1999), 441–454.Google Scholar
[26]Jin, S. and Pareschi, L., Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comp. Phys., 161 (2000), 312–330.CrossRefGoogle Scholar
[27]Jin, S., Pareschi, L. and Toscani, G., Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations, J. Num. Anal., 35 (1998), 2405–2439.Google Scholar
[28]Jin, S., Pareschi, L. and Toscani, G., Uniformly accurate diffusive relaxation schemes for transport equations, J. Num. Anal., 38 (2000), 913–936.Google Scholar
[29]Klar, A., An asymptotic-induced scheme for non stationary transport equations in the diffusive limit, J. Num. Anal., 35 (1998), 1073–1094.Google Scholar
[30]Klar, A., A numerical method for kinetic semiconductor equations in the drift diffusion limit, J. Sci. Comp., 19 (1998), 2032–2050.Google Scholar
[31]Lemou, M. and Mehats, F., Micro-macro schemes for kinetic equations including boundary layers, J. Sci. Comp., 34 (2012), 734–760.Google Scholar
[32]Lemou, M. and Mieussens, L., A new asymptotic preserving scheme Based on micro-macro formulation for linear kinetic equations in the diffusion limit, J. Sci. Comp., 31 (2008), 334–368.Google Scholar
[33]Markowich, P., The Stationary Semiconductor Device Equations, Springer-Verlag, 1986.CrossRefGoogle Scholar
[34]Markowich, P., Poupaud, F. and Schmeiser, C., Diffusion approximation of nonlinear electron phonon collision mechanisms, Mod. Math. et Anal. Num., 29 (1995), 857–869.Google Scholar
[35]Markowich, P., Ringhofer, C. and Schmeiser, C., Semiconductor Equations, Springer-Verlag, Wien-New York, 1989.Google Scholar
[36]Naldi, G. and Pareschi, L., Numerical schemes for kinetic equations in diffusive regimes, Appl. Math. Letters, 11 (1998), 29–35.CrossRefGoogle Scholar
[37]Naldi, G. and Pareschi, L., Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, J. Num. Anal., 37 (2000), 1246–1270.Google Scholar
[38]Pareschi, L. and Russo, G., Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxations, J. Sci. Comp., 25 (2005), 129–155.Google Scholar
[39]Poupaud, F., Diffusion approximationof the linear semiconductor Boltzmann equation: analysis of boundary layers, Asymptotic Anal., 4 (1991), 293–317.Google Scholar
[40]Schmeiser, C. and Zwirchmayr, A., Convergence of moment method for linear kinetic equations, J. Num. Anal., 36 (1998), 74–88.Google Scholar
[41]Shu, C. W., Essentially Non Oscillatory and Weighted Essentially Non Oscillatory schemes for hyperbolic conservation laws, Advanced numerical approximation of nonlinear hyperbolic equations, Lecture Notes in Mathematics, 1697 (2000).Google Scholar