# American Institute of Mathematical Sciences

August  2013, 7(3): 1051-1074. doi: 10.3934/ipi.2013.7.1051

## 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

Received  June 2012 Revised  January 2013 Published  September 2013

For the Wigner equation with discontinuous potentials, a phase space Gaussian beam (PSGB) summation method is proposed in this paper. We first derive the equations satisfied by the parameters for PSGBs and establish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient initial data preparation thus a reduced computational cost than previous method in the literature. The method consists of three steps: 1) Decompose the initial value of the wave function into the sum of GBs and use the parameter relations to prepare the initial values of PSGBs; 2) Solve the evolution equations for each PSGB; 3) Sum all the PSGBs to construct the approximate solution of the Wigner equation. Additionally, in order to connect PSGBs at the discontinuous points of the potential, we provide interface conditions for a single phase space Gaussian beam. Numerical examples are given to verify the validity and accuracy of method.
Citation: Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems and Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
##### 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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##### 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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