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Journal of Scientific Computing
Published in: Journal of Scientific Computing 1/2020

01-10-2020

Optimization Modeling and Simulating of the Stationary Wigner Inflow Boundary Value Problem

Authors: Zhangpeng Sun, Wenqi Yao, Tiao Lu

Published in: Journal of Scientific Computing | Issue 1/2020

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Abstract

The stationary Wigner inflow boundary value problem (SWIBVP) is modeled as an optimization problem by using the idea of shooting method in this paper. To remove the singularity at \(v=0\), we consider a regularized SWIBVP, where a regularization constraint is considered along with the original SWIBVP, and a modified optimization problem is established for it. A shooting algorithm is proposed to solve the two optimization problems, involving the limited-memory BFGS (L-BFGS) algorithm as the optimization solver. Numerical results show that solving the optimization problems with respect to the SWIBVP with the shooting algorithm is as effective as solving the SWIBVP with Frensley’s numerical method (Frensley in Phys Rev B 36:1570–1580, 1987). Furthermore, the modified optimization problem gets rid of the singularity at \(v=0\), and preserves symmetry of the Wigner function, which implies the optimization modeling with respect to the regularized SWIBVP is successful.
previous article On Boundary-Value Problems for RANS Equations and Two-Equation Turbulence Models

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Appendix
Available only for authorised users
Literature
1.
Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40(5), 749–759 (1932)MATHCrossRef
2.
Weinbub, J., Ferry, D.K.: Recent advances in Wigner function approach. Appl. Phys. Rev. 5(4), 041104 (2018)CrossRef
3.
Bardos, C., Caflisch, R.E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Commun. Pure Appl. Math. 39(3), 323–352 (1986)MathSciNetMATHCrossRef
4.
Coron, F.: Computation of the asymptotic states for linear half space kinetic problems. Transp. Theory Stat. Phys. 19(2), 89–114 (1990)MathSciNetMATHCrossRef
5.
Bobylev, A.V., Palczewski, A., Schneider, J.: On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris Sér I(320), 639–644 (1995)MathSciNetMATH
6.
Palczewski, A., Schneider, J., Bobylev, A.V.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM J. Numer. Anal. 34(5), 1865–1883 (1997)MathSciNetMATHCrossRef
7.
Carrillo, J.A., Gamba, I.M., Majorana, A., Shu, C.-W.: A WENO-solver for the transient of devices: performance and comparisons with Monte Carlo methods. J. Comput. Phys. 184, 498–525 (2003)MathSciNetMATHCrossRef
8.
Bernhoff, N.: On half-space problems for the discrete Boltzmann equation. Il nuovo cimento C, Societa italiana di fisica 33(1), 47–54 (2010)MATH
9.
Hu, Z., Li, R., Lu, T., Wang, Y., Yao, W.: Simulation of an \(n^{+}\)-\(n\)-\(n^{+}\) Diode by using globally-hyperbolically-closed high-order moment models. J. Sci. Comput. 59, 761–774 (2014)MathSciNetMATHCrossRef
10.
Li, R., Lu, T., Yao, W.: Discrete kernel preserving model for 1D optical electron–phonon scattering. J. Sci. Comput. 62(2), 317–335 (2015)MathSciNetMATHCrossRef
11.
Li, Q., Lu, J., Sun, W.: A convergent method for linear half-space kinetic equations. ESAIM Math. Model. Numer. Anal. 51(5), 1583–1615 (2017)MathSciNetMATHCrossRef
12.
Yao, W., Lu, T.: Discrete kernel preserving model for 3D electron-optical phonon scattering under arbitrary band structures. J. Sci. Comput. 81(3), 2213–2236 (2019)MathSciNetMATHCrossRef
13.
Frensley, W.R.: Wigner function model of a resonant-tunneling semiconductor device. Phys. Rev. B 36, 1570–1580 (1987)CrossRef
14.
15.
Barletti, L., Zweifel, P.F.: Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions. Transp. Theory Stat. Phys. 30(4–6), 507–520 (2001)MathSciNetMATHCrossRef
16.
Li, R., Lu, T., Sun, Z.P.: Stationary Wigner equation with inflow boundary conditions: will a symmetric potential yield a symmetric solution? SIAM J. Appl. Math. 70(3), 885–897 (2014)MathSciNetMATHCrossRef
17.
Li, R., Lu, T., Sun, Z.P.: Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary condition. Front. Math. China 12(4), 907–919 (2017)MathSciNetMATHCrossRef
18.
Jensen, K.L., Buot, F.A.: Numerical simulation of intrinsic bistability and high-frequency current oscillations in resonant tunneling structures. Phys. Rev. Lett. 66, 1078–1081 (1991)CrossRef
19.
Taj, D., Genovese, L., Rossi, F.: Quantum-transport simulations with the Wigner-function formalism: failure of conventional boundary-condition schemes. Europhys. Lett. 74(6), 1060–1066 (2006)CrossRef
20.
Jiang, H., Lu, T., Cai, W.: A device adaptive inflow boundary condition for Wigner equations of quantum transport. J. Comput. Phys. 258, 773–786 (2014)MathSciNetMATHCrossRef
21.
Dorda, A., Schürrer, F.: A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes. J. Comput. Phys. 284, 95–116 (2015)MathSciNetMATHCrossRef
22.
Shao, S., Lu, T., Cai, W.: Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport. Commun. Comput. Phys. 9(3), 711–739 (2011)MathSciNetMATHCrossRef
23.
24.
Xiong, Y., Chen, Z., Shao, S.: An advective-spectral-mixed method for time-dependent many-body Wigner simulations. SIAM J. Sci. Comput. 38(4), B491–B520 (2016)MathSciNetMATHCrossRef
25.
Chen, Z., Xiong, Y., Shao, S.: Numerical methods for the Wigner equation with unbounded potential. J. Sci. Comput. 79(1), 345–368 (2019)MathSciNetMATHCrossRef
26.
Cai, Z., Fan, Y., Li, R., Lu, T., Wang, Y.: Quantum hydrodynamic model by moment closure of Wigner equation. J. Math. Phys. 53(10), 103503 (2012)MathSciNetMATHCrossRef
27.
Cai, Z., Fan, Y., Li, R., Lu, T., Yao, W.: Quantum hydrodynamic model of density functional theory. J. Math. Chem. 51(7), 1747–1771 (2013)MathSciNetMATHCrossRef
28.
Rossi, F., Jacoboni, C., Nedjalkov, M.: A Monte Carlo solution of the Wigner transport equation. Semicond. Sci. Technol. 9(5S), 934–936 (1994)CrossRef
29.
Shifren, L., Ringhofer, C., Ferry, D.K.: A Wigner function-based quantum ensemble Monte Carlo study of a resonant tunneling diode. IEEE Trans. Electron Dev. 50(3), 769–773 (2003)CrossRef
30.
Stoer, J., Burlisch, R.: Introduction to Numerical Analysis. Springer, New York (1980)CrossRef
31.
Metadata
Title
Optimization Modeling and Simulating of the Stationary Wigner Inflow Boundary Value Problem
Authors
Zhangpeng Sun
Wenqi Yao
Tiao Lu
Publication date
01-10-2020
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2020
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-020-01338-2

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