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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. Degond, Mathematical modelling of microelectronics semiconductor devices,, AMS/IP Studies in Advanced Mathematics, 15 (2000), 77.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[14] |
S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review,, Riv. Mat. Univ. Parma, 3 (2012), 177.
|
[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. |
[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. |
[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. |
[18] |
A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics,, Springer, (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.
doi: 10.1080/13873951003679017. |
[20] |
P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer Verlag Wien, (1990).
doi: 10.1007/978-3-7091-6961-2. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. Degond, Mathematical modelling of microelectronics semiconductor devices,, AMS/IP Studies in Advanced Mathematics, 15 (2000), 77.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[14] |
S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review,, Riv. Mat. Univ. Parma, 3 (2012), 177.
|
[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. |
[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. |
[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. |
[18] |
A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics,, Springer, (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.
doi: 10.1080/13873951003679017. |
[20] |
P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer Verlag Wien, (1990).
doi: 10.1007/978-3-7091-6961-2. |
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