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 & 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

show all references

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

[1]

Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573

[2]

Stéphane Brull, Pierre Degond, Fabrice Deluzet, Alexandre Mouton. Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model. Kinetic & Related Models, 2011, 4 (4) : 991-1023. doi: 10.3934/krm.2011.4.991

[3]

Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic & Related Models, 2011, 4 (4) : 1049-1062. doi: 10.3934/krm.2011.4.1049

[4]

Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044

[5]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[6]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[7]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[8]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65

[9]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[10]

Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789

[11]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[12]

Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019

[13]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641

[14]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105

[15]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[16]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395

[17]

Taposh Kumar Das, Óscar López Pouso. New insights into the numerical solution of the Boltzmann transport equation for photons. Kinetic & Related Models, 2014, 7 (3) : 433-461. doi: 10.3934/krm.2014.7.433

[18]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291

[19]

Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211

[20]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]