# American Institute of Mathematical Sciences

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

 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

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.
Citation: Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic and 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), 3306-3333. doi: 10.1063/1.531567. [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), 205-231. doi: 10.1007/BF02179583. [3] L. Arlotti and M. Lachowicz, Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations, J. Math. Phys., 38 (1997), 3571-3588. doi: 10.1063/1.531869. [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), 55-80. doi: 10.1016/j.jcp.2005.09.005. [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), 3130-3150. doi: 10.1016/j.cma.2009.05.015. [6] P. Degond, Mathematical modelling of microelectronics semiconductor devices, AMS/IP Studies in Advanced Mathematics, AMS and International Press, 15 (2000), 77-110. [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), vol. 135 of The IMA Volumes in Mathematics and its Applications, Springer, New York, (2004), 137-153. doi: 10.1007/978-1-4613-0017-5_8. [8] J. Deng, Asymptotic preserving schemes for semiconductor Boltzmann equation in the diffusive regime, Numer. Math. Theor. Meth. Appl., 5 (2012), 278-296. doi: 10.4208/nmtma.2012.m1045. [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), 1291-1319. [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), 7625-7648. doi: 10.1016/j.jcp.2010.06.017. [11] J. Hu and S. Jin, On kinetic flux vector splitting schemes for quantum Euler equations, Kinet. Relat. Models, 4 (2011), 517-530. doi: 10.3934/krm.2011.4.517. [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), 555-574. doi: 10.1007/s10915-014-9869-2. [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), 806-824. doi: 10.1016/j.jcp.2014.10.050. [14] S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216. [15] S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comput. Phys., 161 (2000), 312-330. doi: 10.1006/jcph.2000.6506. [16] S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913-936. doi: 10.1137/S0036142998347978. [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), B799-B819. doi: 10.1137/120886534. [18] A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. [19] A. Jüngel, Energy transport in semiconductor devices, Math. Computer Modelling Dynam. Sys., 16 (2010), 1-22. doi: 10.1080/13873951003679017. [20] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer Verlag Wien, New York, 1990. doi: 10.1007/978-3-7091-6961-2.

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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), 3306-3333. doi: 10.1063/1.531567. [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), 205-231. doi: 10.1007/BF02179583. [3] L. Arlotti and M. Lachowicz, Euler and Navier-Stokes limits of the Uehling-Uhlenbeck quantum kinetic equations, J. Math. Phys., 38 (1997), 3571-3588. doi: 10.1063/1.531869. [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), 55-80. doi: 10.1016/j.jcp.2005.09.005. [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), 3130-3150. doi: 10.1016/j.cma.2009.05.015. [6] P. Degond, Mathematical modelling of microelectronics semiconductor devices, AMS/IP Studies in Advanced Mathematics, AMS and International Press, 15 (2000), 77-110. [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), vol. 135 of The IMA Volumes in Mathematics and its Applications, Springer, New York, (2004), 137-153. doi: 10.1007/978-1-4613-0017-5_8. [8] J. Deng, Asymptotic preserving schemes for semiconductor Boltzmann equation in the diffusive regime, Numer. Math. Theor. Meth. Appl., 5 (2012), 278-296. doi: 10.4208/nmtma.2012.m1045. [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), 1291-1319. [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), 7625-7648. doi: 10.1016/j.jcp.2010.06.017. [11] J. Hu and S. Jin, On kinetic flux vector splitting schemes for quantum Euler equations, Kinet. Relat. Models, 4 (2011), 517-530. doi: 10.3934/krm.2011.4.517. [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), 555-574. doi: 10.1007/s10915-014-9869-2. [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), 806-824. doi: 10.1016/j.jcp.2014.10.050. [14] S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216. [15] S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comput. Phys., 161 (2000), 312-330. doi: 10.1006/jcph.2000.6506. [16] S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913-936. doi: 10.1137/S0036142998347978. [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), B799-B819. doi: 10.1137/120886534. [18] A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8. [19] A. Jüngel, Energy transport in semiconductor devices, Math. Computer Modelling Dynam. Sys., 16 (2010), 1-22. doi: 10.1080/13873951003679017. [20] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer Verlag Wien, New York, 1990. doi: 10.1007/978-3-7091-6961-2.
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