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A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
A random cloud model for the Wigner equation
1. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39 - 10117 Berlin |
References:
[1] |
L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass., 1968. |
[2] |
M. H. A. Davis, Markov Models and Optimization, Chapman & Hall, London, 1993.
doi: 10.1007/978-1-4899-4483-2. |
[3] |
I. M. Gamba, M. P. Gualdani and R. W. Sharp, An adaptable discontinuous Galerkin scheme for the Wigner-Fokker-Planck equation, Commun. Math. Sci., 7 (2009), 635-664, URL http://projecteuclid.org/euclid.cms/1256562817.
doi: 10.4310/CMS.2009.v7.n3.a7. |
[4] |
I. M. Gamba, S. Rjasanow and W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Modelling, 42 (2005), 683-700.
doi: 10.1016/j.mcm.2004.02.047. |
[5] |
T. M. M. Homolle and N. G. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comput. Phys., 226 (2007), 2341-2358.
doi: 10.1016/j.jcp.2007.07.006. |
[6] |
P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Methods Appl. Sci., 11 (1989), 459-469.
doi: 10.1002/mma.1670110404. |
[7] |
P. A. Markowich and C. A. Ringhofer, An analysis of the quantum Liouville equation, Z. Angew. Math. Mech., 69 (1989), 121-127.
doi: 10.1002/zamm.19890690303. |
[8] |
M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer and D. K. Ferry, Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices, Phys. Rev. B, 70 (2004), 115319.
doi: 10.1103/PhysRevB.70.115319. |
[9] |
R. I. A. Patterson and W. Wagner, A stochastic weighted particle method for coagulation-advection problems, SIAM J. Sci. Comput., 34 (2012), B290-B311.
doi: 10.1137/110843319. |
[10] |
R. I. A. Patterson, W. Wagner and M. Kraft, Stochastic weighted particle methods for population balance equations, J. Comput. Phys., 230 (2011), 7456-7472.
doi: 10.1016/j.jcp.2011.06.011. |
[11] |
D. Querlioz and P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices, Wiley, 2010.
doi: 10.1002/9781118618479. |
[12] |
G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Phys. Fluids, 23 (2011), 030606(1-12).
doi: 10.1063/1.3558887. |
[13] |
S. Rjasanow and W. Wagner, A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), 243-253.
doi: 10.1006/jcph.1996.0057. |
[14] |
S. Rjasanow and W. Wagner, Simulation of rare events by the stochastic weighted particle method for the Boltzmann equation, Math. Comput. Modelling, 33 (2001), 907-926.
doi: 10.1016/S0895-7177(00)00289-2. |
[15] |
J. M. Sellier, M. Nedjalkov, I. Dimov and S. Selberherr, A benchmark study of the Wigner Monte Carlo method, Monte Carlo Methods Appl., 20 (2014), 43-51.
doi: 10.1515/mcma-2013-0018. |
[16] |
W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation, Monte Carlo Methods Appl., 14 (2008), 191-268.
doi: 10.1515/MCMA.2008.010. |
[17] |
W. Wagner, A random cloud model for the Schrödinger equation, Kinetic and Related Models, 7 (2014), 361-379.
doi: 10.3934/krm.2014.7.361. |
[18] |
E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. |
show all references
References:
[1] |
L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass., 1968. |
[2] |
M. H. A. Davis, Markov Models and Optimization, Chapman & Hall, London, 1993.
doi: 10.1007/978-1-4899-4483-2. |
[3] |
I. M. Gamba, M. P. Gualdani and R. W. Sharp, An adaptable discontinuous Galerkin scheme for the Wigner-Fokker-Planck equation, Commun. Math. Sci., 7 (2009), 635-664, URL http://projecteuclid.org/euclid.cms/1256562817.
doi: 10.4310/CMS.2009.v7.n3.a7. |
[4] |
I. M. Gamba, S. Rjasanow and W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Modelling, 42 (2005), 683-700.
doi: 10.1016/j.mcm.2004.02.047. |
[5] |
T. M. M. Homolle and N. G. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comput. Phys., 226 (2007), 2341-2358.
doi: 10.1016/j.jcp.2007.07.006. |
[6] |
P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Methods Appl. Sci., 11 (1989), 459-469.
doi: 10.1002/mma.1670110404. |
[7] |
P. A. Markowich and C. A. Ringhofer, An analysis of the quantum Liouville equation, Z. Angew. Math. Mech., 69 (1989), 121-127.
doi: 10.1002/zamm.19890690303. |
[8] |
M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer and D. K. Ferry, Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices, Phys. Rev. B, 70 (2004), 115319.
doi: 10.1103/PhysRevB.70.115319. |
[9] |
R. I. A. Patterson and W. Wagner, A stochastic weighted particle method for coagulation-advection problems, SIAM J. Sci. Comput., 34 (2012), B290-B311.
doi: 10.1137/110843319. |
[10] |
R. I. A. Patterson, W. Wagner and M. Kraft, Stochastic weighted particle methods for population balance equations, J. Comput. Phys., 230 (2011), 7456-7472.
doi: 10.1016/j.jcp.2011.06.011. |
[11] |
D. Querlioz and P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices, Wiley, 2010.
doi: 10.1002/9781118618479. |
[12] |
G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Phys. Fluids, 23 (2011), 030606(1-12).
doi: 10.1063/1.3558887. |
[13] |
S. Rjasanow and W. Wagner, A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), 243-253.
doi: 10.1006/jcph.1996.0057. |
[14] |
S. Rjasanow and W. Wagner, Simulation of rare events by the stochastic weighted particle method for the Boltzmann equation, Math. Comput. Modelling, 33 (2001), 907-926.
doi: 10.1016/S0895-7177(00)00289-2. |
[15] |
J. M. Sellier, M. Nedjalkov, I. Dimov and S. Selberherr, A benchmark study of the Wigner Monte Carlo method, Monte Carlo Methods Appl., 20 (2014), 43-51.
doi: 10.1515/mcma-2013-0018. |
[16] |
W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation, Monte Carlo Methods Appl., 14 (2008), 191-268.
doi: 10.1515/MCMA.2008.010. |
[17] |
W. Wagner, A random cloud model for the Schrödinger equation, Kinetic and Related Models, 7 (2014), 361-379.
doi: 10.3934/krm.2014.7.361. |
[18] |
E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. |
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