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The Gaussian beam method for the wigner equation with discontinuous potentials
1. | Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 |
2. | Department of Mathematics, Institute of Nature Science, and Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China |
3. | Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240 |
References:
[1] |
J. Akian, R. Alexandre and S. Bougacha, A Gaussian beam approach for computing Wigner measures in convex domains, Kinetic and Related Models, 4 (2011), 589-631.
doi: 10.3934/krm.2011.4.589. |
[2] |
G. Ariel, B. Engquist, N. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization, Journal of Computational Physics, 230 (2011), 2303-2321. |
[3] |
A. Arnold and F. Nier, Numerical analysis of the deterministic particle method applied to the wigner equation, Mathematics of Computation, 58 (1992), 645-669.
doi: 10.1090/S0025-5718-1992-1122055-5. |
[4] |
A. Arnold and C. Ringhofer, Operator splitting methods applied to spectral discretizations of quantum transport equations, SIAM J. Numer. Anal., 32 (1995), 1876-1894.
doi: 10.1137/0732084. |
[5] |
A. Arnold, H. Lange and P. Zweifel, A discrete-velocity, stationary Wigner equation, J. Math. Phys., 41 (2000), 7167-7180.
doi: 10.1063/1.1318732. |
[6] |
A. Arnold, Mathematical properties of quantum evolution equations, In "Quantum Transport-Modelling, Analysis and Asymptotics" (eds. G. Allaire, A. Arnold, P. Degond, and T. Hou), Lecture Notes Math. Springer, Berlin, 1946 (2008), 45-110. |
[7] |
W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.
doi: 10.1006/jcph.2001.6956. |
[8] |
S. Bougacha, J. Akian and R. Alexandre, Gaussian beam summation for the wave equation in a convex domain, Commun. Math. Sci., 7 (2009), 973-1008. |
[9] |
L. Dieci and T. Eirola, Positive definiteness in the numerical solution of Riccati differential equations, Numer. Math., 67 (1994), 303-313.
doi: 10.1007/s002110050030. |
[10] |
W. Frensley, Wigner-function model of a resonant-tunneling semiconductor device, Phys. Rev. B, 36 (1987), 1570-1580.
doi: 10.1103/PhysRevB.36.1570. |
[11] |
P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. |
[12] |
T. Goudon, Analysis of a semidiscrete version of theWigner equation, SIAM J. Numer. Anal., 40 (2002), 2007-2025.
doi: 10.1137/S0036142901388366. |
[13] |
H. Guo and R. Chen, Short-time Chebyshev propagator for the Liouville-von Neumann equation, Journal of Chemical Physics, 110 (1999), 6626-6623.
doi: 10.1063/1.478570. |
[14] |
E. J. Heller, Time-dependent approach to semiclassical dynamics, The Journal of Chemical Physics, 62 (1975), 1544-1555.
doi: 10.1063/1.430620. |
[15] |
I. Horenko, B. Schmidt and C. Schütte, Multdimensional classical Liouville dynamics with quantum initial conditions, Journal of Chermical Physics, 117 (2002), 4643-4650.
doi: 10.1063/1.1498467. |
[16] |
N. Kluksdahl, A. Kriman, D. Ferry and C. Ringhofer, Self-consistent study of the resonant tunneling diode, Phys. Rev. B., 39 (1989), 7720-7735.
doi: 10.1103/PhysRevB.39.7720. |
[17] |
S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numerica, 20 (2011), 121-209.
doi: 10.1017/S0962492911000031. |
[18] |
S. Jin, D. Wei and D. Yin, Gaussian beam methods for the Schrödinger equation with discontinuous potentials,, submit to Journal of Computational and Applied Mathematics., ().
|
[19] |
S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Communications in Mathematical Sciences, 6 (2008), 995-1020. |
[20] |
H. Kosina and M. Nedjalkov, Wigner function-based device modeling, In "Handbook of Theoretical and Computational Nanotechnology" (eds. M. Rieth and W. Schommers), American Scientific Publishers, Los Angeles, 10 (2006), 731-763. |
[21] |
S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, Journal of Computational Physics, 228 (2009), 2951-2977.
doi: 10.1016/j.jcp.2009.01.007. |
[22] |
S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high-frequency wave propagation, Geophysics, 72 (2007), 61-76.
doi: 10.1190/1.2752136. |
[23] |
P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.
doi: 10.4171/RMI/143. |
[24] |
H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions, Math. Comp., 82 (2013), 919-952. |
[25] |
P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[26] |
P. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Meth. Appl. Sci., 11 (1989), 459-469.
doi: 10.1002/mma.1670110404. |
[27] |
M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave motion, 47 (2010), 421-439.
doi: 10.1016/j.wavemoti.2010.02.001. |
[28] |
M. Nedjalkov, R. Kosik, H. Kosina and S. Selberherr, Wigner transport through tunneling structures-scattering interpretation of the potential operator, In" International Conference on Simulation of Semiconductor Processes and Devices SISPAD" (2002), 187-190.
doi: 10.1109/SISPAD.2002.1034548. |
[29] |
M. Nedjalkov, D. Vasileska, D. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov and V. Palanovski, Wigner transport models of the electron-phonon kinetics in quantum wires, Phys. Rev. B., 74 (2006), 035311.
doi: 10.1103/PhysRevB.74.035311. |
[30] |
A. Norris, S. White and J. Schrieffer, Gaussian wave packets in inhomogeneous media with curved interfaces, Pro. R. Soc. Lond. A, 412 (1987), 93-123.
doi: 10.1098/rspa.1987.0082. |
[31] |
M. M. Popov, A new method of computation of wave fields using Gaussian beams, Wave Motion, 4 (1982), 85-97.
doi: 10.1016/0165-2125(82)90016-6. |
[32] |
J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schröinger equation, J. Comp. Phys., 229 (2010), 7848-7873.
doi: 10.1016/j.jcp.2010.06.043. |
[33] |
J. Ralston, Gaussian beams and the propagation of singularities, Studies in PDEs, MAA stud. Math., 23 (1982), 206-248. |
[34] |
U. Ravaioli, M. Osman, W. Pötz, N. Kluksdahl and D. Ferry, Investigation of ballistic transport through resonant-tunnelling quantum wells using Wigner function approach, Physica B., 134 (1985), 36-40.
doi: 10.1016/0378-4363(85)90317-1. |
[35] |
C. Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM J. Numer. Anal., 27 (1990), 32-50.
doi: 10.1137/0727003. |
[36] |
C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices, Acta Numerica., 6 (1997), 485-521.
doi: 10.1017/S0962492900002762. |
[37] |
D. Robert, On the Herman-Kluk seimiclassical approximation, Reviews in Mathematical Physics, 22 (2010), 1123-1145.
doi: 10.1142/S0129055X1000417X. |
[38] |
L. Shifren and D. Ferry, A Wigner function based ensemble Monte Carlo approach for accurate incorporation of quantum effects in device simulation, J. Comp. Electr., 1 (2002), 55-58. |
[39] |
V. Sverdlov, E. Ungersboeck, H. Kosina and S. Selberherr, Current transport models for nanoscale semiconductor devices, Materials Sci. Engin. R., 58 (2008), 228-270.
doi: 10.1016/j.mser.2007.11.001. |
[40] |
N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228 (2009), 8856-8871.
doi: 10.1016/j.jcp.2009.08.028. |
[41] |
N. M. Tanushev, J. L. Qian and J. Ralston, Mountain waves and Gaussian beams, SIAM Multiscale Modeling and Simulation, 6 (2007), 688-709.
doi: 10.1137/060673667. |
[42] |
N. M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475. |
[43] |
E. Wigner, On the quantum correction for the thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. |
[44] |
D. Yin and C. Zheng, Gaussian beam formulation and interface conditions for the one-dimensional linear Schödinger equation, Wave Motion, 48 (2011), 310-324.
doi: 10.1016/j.wavemoti.2010.11.006. |
[45] |
D. Yin and C. Zheng, Composite coherent states approximation for one-dimensional multi-phased wave functions, Communications in Computational Physics, 11 (2012), 951-984.
doi: 10.4208/cicp.101010.250511a. |
show all references
References:
[1] |
J. Akian, R. Alexandre and S. Bougacha, A Gaussian beam approach for computing Wigner measures in convex domains, Kinetic and Related Models, 4 (2011), 589-631.
doi: 10.3934/krm.2011.4.589. |
[2] |
G. Ariel, B. Engquist, N. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization, Journal of Computational Physics, 230 (2011), 2303-2321. |
[3] |
A. Arnold and F. Nier, Numerical analysis of the deterministic particle method applied to the wigner equation, Mathematics of Computation, 58 (1992), 645-669.
doi: 10.1090/S0025-5718-1992-1122055-5. |
[4] |
A. Arnold and C. Ringhofer, Operator splitting methods applied to spectral discretizations of quantum transport equations, SIAM J. Numer. Anal., 32 (1995), 1876-1894.
doi: 10.1137/0732084. |
[5] |
A. Arnold, H. Lange and P. Zweifel, A discrete-velocity, stationary Wigner equation, J. Math. Phys., 41 (2000), 7167-7180.
doi: 10.1063/1.1318732. |
[6] |
A. Arnold, Mathematical properties of quantum evolution equations, In "Quantum Transport-Modelling, Analysis and Asymptotics" (eds. G. Allaire, A. Arnold, P. Degond, and T. Hou), Lecture Notes Math. Springer, Berlin, 1946 (2008), 45-110. |
[7] |
W. Bao, S. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.
doi: 10.1006/jcph.2001.6956. |
[8] |
S. Bougacha, J. Akian and R. Alexandre, Gaussian beam summation for the wave equation in a convex domain, Commun. Math. Sci., 7 (2009), 973-1008. |
[9] |
L. Dieci and T. Eirola, Positive definiteness in the numerical solution of Riccati differential equations, Numer. Math., 67 (1994), 303-313.
doi: 10.1007/s002110050030. |
[10] |
W. Frensley, Wigner-function model of a resonant-tunneling semiconductor device, Phys. Rev. B, 36 (1987), 1570-1580.
doi: 10.1103/PhysRevB.36.1570. |
[11] |
P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. |
[12] |
T. Goudon, Analysis of a semidiscrete version of theWigner equation, SIAM J. Numer. Anal., 40 (2002), 2007-2025.
doi: 10.1137/S0036142901388366. |
[13] |
H. Guo and R. Chen, Short-time Chebyshev propagator for the Liouville-von Neumann equation, Journal of Chemical Physics, 110 (1999), 6626-6623.
doi: 10.1063/1.478570. |
[14] |
E. J. Heller, Time-dependent approach to semiclassical dynamics, The Journal of Chemical Physics, 62 (1975), 1544-1555.
doi: 10.1063/1.430620. |
[15] |
I. Horenko, B. Schmidt and C. Schütte, Multdimensional classical Liouville dynamics with quantum initial conditions, Journal of Chermical Physics, 117 (2002), 4643-4650.
doi: 10.1063/1.1498467. |
[16] |
N. Kluksdahl, A. Kriman, D. Ferry and C. Ringhofer, Self-consistent study of the resonant tunneling diode, Phys. Rev. B., 39 (1989), 7720-7735.
doi: 10.1103/PhysRevB.39.7720. |
[17] |
S. Jin, P. Markowich and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numerica, 20 (2011), 121-209.
doi: 10.1017/S0962492911000031. |
[18] |
S. Jin, D. Wei and D. Yin, Gaussian beam methods for the Schrödinger equation with discontinuous potentials,, submit to Journal of Computational and Applied Mathematics., ().
|
[19] |
S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Communications in Mathematical Sciences, 6 (2008), 995-1020. |
[20] |
H. Kosina and M. Nedjalkov, Wigner function-based device modeling, In "Handbook of Theoretical and Computational Nanotechnology" (eds. M. Rieth and W. Schommers), American Scientific Publishers, Los Angeles, 10 (2006), 731-763. |
[21] |
S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, Journal of Computational Physics, 228 (2009), 2951-2977.
doi: 10.1016/j.jcp.2009.01.007. |
[22] |
S. Leung, J. Qian and R. Burridge, Eulerian Gaussian beams for high-frequency wave propagation, Geophysics, 72 (2007), 61-76.
doi: 10.1190/1.2752136. |
[23] |
P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.
doi: 10.4171/RMI/143. |
[24] |
H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions, Math. Comp., 82 (2013), 919-952. |
[25] |
P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[26] |
P. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Meth. Appl. Sci., 11 (1989), 459-469.
doi: 10.1002/mma.1670110404. |
[27] |
M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave motion, 47 (2010), 421-439.
doi: 10.1016/j.wavemoti.2010.02.001. |
[28] |
M. Nedjalkov, R. Kosik, H. Kosina and S. Selberherr, Wigner transport through tunneling structures-scattering interpretation of the potential operator, In" International Conference on Simulation of Semiconductor Processes and Devices SISPAD" (2002), 187-190.
doi: 10.1109/SISPAD.2002.1034548. |
[29] |
M. Nedjalkov, D. Vasileska, D. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov and V. Palanovski, Wigner transport models of the electron-phonon kinetics in quantum wires, Phys. Rev. B., 74 (2006), 035311.
doi: 10.1103/PhysRevB.74.035311. |
[30] |
A. Norris, S. White and J. Schrieffer, Gaussian wave packets in inhomogeneous media with curved interfaces, Pro. R. Soc. Lond. A, 412 (1987), 93-123.
doi: 10.1098/rspa.1987.0082. |
[31] |
M. M. Popov, A new method of computation of wave fields using Gaussian beams, Wave Motion, 4 (1982), 85-97.
doi: 10.1016/0165-2125(82)90016-6. |
[32] |
J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schröinger equation, J. Comp. Phys., 229 (2010), 7848-7873.
doi: 10.1016/j.jcp.2010.06.043. |
[33] |
J. Ralston, Gaussian beams and the propagation of singularities, Studies in PDEs, MAA stud. Math., 23 (1982), 206-248. |
[34] |
U. Ravaioli, M. Osman, W. Pötz, N. Kluksdahl and D. Ferry, Investigation of ballistic transport through resonant-tunnelling quantum wells using Wigner function approach, Physica B., 134 (1985), 36-40.
doi: 10.1016/0378-4363(85)90317-1. |
[35] |
C. Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM J. Numer. Anal., 27 (1990), 32-50.
doi: 10.1137/0727003. |
[36] |
C. Ringhofer, Computational methods for semiclassical and quantum transport in semiconductor devices, Acta Numerica., 6 (1997), 485-521.
doi: 10.1017/S0962492900002762. |
[37] |
D. Robert, On the Herman-Kluk seimiclassical approximation, Reviews in Mathematical Physics, 22 (2010), 1123-1145.
doi: 10.1142/S0129055X1000417X. |
[38] |
L. Shifren and D. Ferry, A Wigner function based ensemble Monte Carlo approach for accurate incorporation of quantum effects in device simulation, J. Comp. Electr., 1 (2002), 55-58. |
[39] |
V. Sverdlov, E. Ungersboeck, H. Kosina and S. Selberherr, Current transport models for nanoscale semiconductor devices, Materials Sci. Engin. R., 58 (2008), 228-270.
doi: 10.1016/j.mser.2007.11.001. |
[40] |
N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228 (2009), 8856-8871.
doi: 10.1016/j.jcp.2009.08.028. |
[41] |
N. M. Tanushev, J. L. Qian and J. Ralston, Mountain waves and Gaussian beams, SIAM Multiscale Modeling and Simulation, 6 (2007), 688-709.
doi: 10.1137/060673667. |
[42] |
N. M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475. |
[43] |
E. Wigner, On the quantum correction for the thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. |
[44] |
D. Yin and C. Zheng, Gaussian beam formulation and interface conditions for the one-dimensional linear Schödinger equation, Wave Motion, 48 (2011), 310-324.
doi: 10.1016/j.wavemoti.2010.11.006. |
[45] |
D. Yin and C. Zheng, Composite coherent states approximation for one-dimensional multi-phased wave functions, Communications in Computational Physics, 11 (2012), 951-984.
doi: 10.4208/cicp.101010.250511a. |
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