An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach

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December 2015, 8(4): 707-723. doi: 10.3934/krm.2015.8.707

An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach

Jingwei Hu ^1, , Shi Jin ^2, and Li Wang ^3,

1.	Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, United States
2.	Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706
3.	Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, United States

Received August 2014 Revised April 2015 Published July 2015

Abstract
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We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision [10] cannot capture the correct energy-transport limit. This problem was addressed in [13], where a thresholded BGK penalization was introduced. Here we propose an alternative based on a splitting approach. It has the advantage of treating the collisions at different scales separately, hence is free of choosing threshold and easier to implement. Formal asymptotic analysis and numerical results validate the efficiency and accuracy of the proposed scheme.

Keywords: Semiconductor Boltzmann equation, asymptotic-preserving scheme, splitting method., energy-transport system.

Mathematics Subject Classification: Primary: 65M06, 35L45, 35K15, 65Z0.

Citation: Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707

References:

[1]	N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567. Google Scholar
[2]	N. B. Abdallah, P. Degond and S. Genieys, An energy-transport model for semiconductors derived from the Boltzmann equation,, J. Stat. Phys., 84 (1996), 205. doi: 10.1007/BF02179583. Google Scholar
[3]	L. Arlotti and M. Lachowicz, Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations,, J. Math. Phys., 38 (1997), 3571. doi: 10.1063/1.531869. Google Scholar
[4]	J. Carrillo, I. Gamba, A. Majorana and C.-W. Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods,, J. Comput. Phys., 214 (2006), 55. doi: 10.1016/j.jcp.2005.09.005. Google Scholar
[5]	Y. Cheng, I. Gamba, A. Majorana and C.-W. Shu, A discontinous Galerkin solver for Boltzmann-Poisson systems in nano devices,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3130. doi: 10.1016/j.cma.2009.05.015. Google Scholar
[6]	P. Degond, Mathematical modelling of microelectronics semiconductor devices,, AMS/IP Studies in Advanced Mathematics, 15 (2000), 77. Google Scholar
[7]	P. Degond, C. D. Levermore and C. Schmeiser, A note on the Energy-Transport limit of the semiconductor Boltzmann equation,, in Transport in Transition Regimes* (ed. N. B. A. et al)*, (2004), 137. doi: 10.1007/978-1-4613-0017-5_8. Google Scholar
[8]	J. Deng, Asymptotic preserving schemes for semiconductor Boltzmann equation in the diffusive regime,, Numer. Math. Theor. Meth. Appl., 5 (2012), 278. doi: 10.4208/nmtma.2012.m1045. Google Scholar
[9]	G. Dimarco, L. Pareschi and V. Rispoli, Implicit-explicit Runge-Kutta schemes for the Boltzmann-Poisson system for semiconductors,, Commun. Comput. Phys., 15 (2014), 1291. Google Scholar
[10]	F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources,, J. Comput. Phys., 229 (2010), 7625. doi: 10.1016/j.jcp.2010.06.017. Google Scholar
[11]	J. Hu and S. Jin, On kinetic flux vector splitting schemes for quantum Euler equations,, Kinet. Relat. Models, 4 (2011), 517. doi: 10.3934/krm.2011.4.517. Google Scholar
[12]	J. Hu, Q. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy,, J. Sci. Comput., 62 (2015), 555. doi: 10.1007/s10915-014-9869-2. Google Scholar
[13]	J. Hu and L. Wang, An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit,, J. Comput. Phys., 281 (2015), 806. doi: 10.1016/j.jcp.2014.10.050. Google Scholar
[14]	S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review,, Riv. Mat. Univ. Parma, 3 (2012), 177. Google Scholar
[15]	S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes,, J. Comput. Phys., 161 (2000), 312. doi: 10.1006/jcph.2000.6506. Google Scholar
[16]	S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations,, SIAM J. Numer. Anal., 38 (2000), 913. doi: 10.1137/S0036142998347978. Google Scholar
[17]	S. Jin and L. Wang, Asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/120886534. Google Scholar
[18]	A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics,, Springer, (2009). doi: 10.1007/978-3-540-89526-8. Google Scholar
[19]	A. Jüngel, Energy transport in semiconductor devices,, Math. Computer Modelling Dynam. Sys., 16 (2010), 1. doi: 10.1080/13873951003679017. Google Scholar
[20]	P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer Verlag Wien, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

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References:

[1]	N. B. Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567. Google Scholar
[2]	N. B. Abdallah, P. Degond and S. Genieys, An energy-transport model for semiconductors derived from the Boltzmann equation,, J. Stat. Phys., 84 (1996), 205. doi: 10.1007/BF02179583. Google Scholar
[3]	L. Arlotti and M. Lachowicz, Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations,, J. Math. Phys., 38 (1997), 3571. doi: 10.1063/1.531869. Google Scholar
[4]	J. Carrillo, I. Gamba, A. Majorana and C.-W. Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods,, J. Comput. Phys., 214 (2006), 55. doi: 10.1016/j.jcp.2005.09.005. Google Scholar
[5]	Y. Cheng, I. Gamba, A. Majorana and C.-W. Shu, A discontinous Galerkin solver for Boltzmann-Poisson systems in nano devices,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3130. doi: 10.1016/j.cma.2009.05.015. Google Scholar
[6]	P. Degond, Mathematical modelling of microelectronics semiconductor devices,, AMS/IP Studies in Advanced Mathematics, 15 (2000), 77. Google Scholar
[7]	P. Degond, C. D. Levermore and C. Schmeiser, A note on the Energy-Transport limit of the semiconductor Boltzmann equation,, in Transport in Transition Regimes* (ed. N. B. A. et al)*, (2004), 137. doi: 10.1007/978-1-4613-0017-5_8. Google Scholar
[8]	J. Deng, Asymptotic preserving schemes for semiconductor Boltzmann equation in the diffusive regime,, Numer. Math. Theor. Meth. Appl., 5 (2012), 278. doi: 10.4208/nmtma.2012.m1045. Google Scholar
[9]	G. Dimarco, L. Pareschi and V. Rispoli, Implicit-explicit Runge-Kutta schemes for the Boltzmann-Poisson system for semiconductors,, Commun. Comput. Phys., 15 (2014), 1291. Google Scholar
[10]	F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources,, J. Comput. Phys., 229 (2010), 7625. doi: 10.1016/j.jcp.2010.06.017. Google Scholar
[11]	J. Hu and S. Jin, On kinetic flux vector splitting schemes for quantum Euler equations,, Kinet. Relat. Models, 4 (2011), 517. doi: 10.3934/krm.2011.4.517. Google Scholar
[12]	J. Hu, Q. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy,, J. Sci. Comput., 62 (2015), 555. doi: 10.1007/s10915-014-9869-2. Google Scholar
[13]	J. Hu and L. Wang, An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit,, J. Comput. Phys., 281 (2015), 806. doi: 10.1016/j.jcp.2014.10.050. Google Scholar
[14]	S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review,, Riv. Mat. Univ. Parma, 3 (2012), 177. Google Scholar
[15]	S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes,, J. Comput. Phys., 161 (2000), 312. doi: 10.1006/jcph.2000.6506. Google Scholar
[16]	S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations,, SIAM J. Numer. Anal., 38 (2000), 913. doi: 10.1137/S0036142998347978. Google Scholar
[17]	S. Jin and L. Wang, Asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime,, SIAM J. Sci. Comput., 35 (2013). doi: 10.1137/120886534. Google Scholar
[18]	A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics,, Springer, (2009). doi: 10.1007/978-3-540-89526-8. Google Scholar
[19]	A. Jüngel, Energy transport in semiconductor devices,, Math. Computer Modelling Dynam. Sys., 16 (2010), 1. doi: 10.1080/13873951003679017. Google Scholar
[20]	P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer Verlag Wien, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

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