JCM
Journal of Computational Mathematics ›› 2021, Vol. 39 ›› Issue (1): 22-42.DOI: 10.4208/jcm.1906-m2018-0081

Previous Articles     Next Articles

A HYBRID EXPLICIT-IMPLICIT SCHEME FOR THE TIME-DEPENDENT WIGNER EQUATION

Haiyan Jiang1, Tiao Lu2, Xu Yin1   

  1. 1. Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    2. CAPT, HEDPS, LMAM, IFSA Collaborative Innovation Center of MoE, School of Mathematical Sciences, Peking University, Beijing 100871, China
  • Received:2018-05-01 Online:2021-01-15 Published:2021-03-11
  • Supported by:
    This research was supported in part by the NSFC (91434201, 91630130, 11671038, 11421101).
PDF ( 9 ) Cited

Haiyan Jiang, Tiao Lu, Xu Yin. A HYBRID EXPLICIT-IMPLICIT SCHEME FOR THE TIME-DEPENDENT WIGNER EQUATION[J]. Journal of Computational Mathematics, 2021, 39(1): 22-42.

This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation, and gives the error analysis for the full discretization of its initial value problem. An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to discretize the advection term. The consistence error and the stability of the full discretization are analyzed. A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail. The final convergence result shows clearly how the regularity of the solution affects the convergence order of the proposed scheme. Numerical results are presented for confirming the sharpness of the analysis. The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated. The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.
Key words: Finite difference method, Spectral method, Hybrid scheme, Error analysis, Wigner equation

CLC Number: 

[1] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40:5(1932), 749-759.
[2] D. Ferry and S. Goodnick, Transport in Nanostructures, Cambridge Univ. Press, Cambridge, U.K, 1997.
[3] J. Shi and I. Gamba, A high order local solver for Wigner equation, International Workshop on Computational Electronics, (2004), 245-246.
[4] W. Schleich, Quantum Optics in Phase Space, Wiley, England, 2001.
[5] W. Frensley,Wigner function model of a resonant-tunneling semiconductor device, Phys. Rev. B, 36(1987), 1570-1580.
[6] K. Jensen and F. Buot, Numerical simulation of transient response and resonant tunneling characteristics of double barrier semiconductor structures as a function of experimental parameters, J. Appl. Phys., 65:12(1989), 5248-5250.
[7] K. Jensen and F. Buot, Numerical aspects on the simulation of IV characteristics and switching times of resonant tunneling diodes, J. Appl. Phys., 67(1990), 2153-2155.
[8] A. Dorda and F. Schürrer, A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes, J. Comput. Phys., 284(2015), 95-116.
[9] C. Ringhofer, A spectral method for the numerical solution of quantum tunneling phenomena, SIAM J. Num. Anal., 27(1990), 32-50.
[10] C. Ringhofer, A spectral collocation technique for the solution of the Wigner-Poisson problem, SIAM Journal on Numerical Analysis, 29:3(1992), 679-700.
[11] K. Kim and B. Lee, On the high order numerical calculation schemes for the Wigner transport equation, Solid-State Electronics, 43:12(1999), 2243-2245.
[12] D. Yin, M. Tang, and S. Jin, The Gaussian beam method for the Wigner equation with discontinuous potentials, Inverse Problems & Imaging, 7:3(2013).
[13] Z. Cai, Y. Fan, R. Li, T. Lu, and Y. Wang, Quantum hydrodynamics models by moment closure of Wigner equation, J. Math. Phys., 53:103503(2012).
[14] R. Li, T. Lu, Y. Wang, and W. Yao, Numerical method for high order hyperbolic moment system of Wigner equation, Commun. Comput. Phys., 9:3(2014), 659-698.
[15] S. Shao, T. Lu, and W. Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Commun. Comput. Phys., 9(2011), 711-739.
[16] P. Markowich and C. Ringhofer, An analysis of the quantum liouville equation, Z. angew. Math. Mech., 69(1989), 121-127.
[17] H. Jiang and W. Cai, Effect of boundary treatments on quantum transport current in the Green's function and Wigner distribution methods for a nano-scale DG-MOSFET, J. Comput. Phys., 229:12(2010), 4461-4475.
[18] H. Jiang, W. Cai, and R. Tsu, Accuracy of the frensley inflow boundary condition for Wigner equations in simulating resonant tunneling diodes, J. Comput. Phys., 230(2011), 2031-2044.
[19] H. Jiang, T. Lu, and W. Cai, A device adaptive inflow boundary condition for Wigner equations of quantum transport, J. Comput. Phys., 248(2014), 773-786.
[20] A. Arnold, H. Lange, and P. Zweifel, A discrete-velocity, stationary Wigner equation, J. Math. Phys., 41:11(2000), 7167-7180.
[21] T. Goudon, Analysis of a semidiscrete version of the Wigner equation, SIAM J. Numerical Analysis, 40:6(2003), 2007-2025.
[22] R. Li, T. Lu, and Z.P. Sun, Stationary wigner equation with inflow boundary conditions:Will a symmetric potential yield a symmetric solution?, SIAM J. Appl. Math., 70:3(2014), 885-897.
[23] R. Li, T. Lu, and Z.P. Sun, Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions, Frontiers of Mathematics in China, 12:4(2017), 907-919.
[24] R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations:SteadyState and Time-Dependent Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2007.
[1] Baiying Dong, Xiufeng Feng, Zhilin Li. AN L∞ SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(6): 882-912.
[2] Libin Liu, Yanping Chen, Ying Liang. NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID [J]. Journal of Computational Mathematics, 2022, 40(2): 258-274.
[3] Pengzhan Huang, Yinnian He, Ting Li. A FINITE ELEMENT ALGORITHM FOR NEMATIC LIQUID CRYSTAL FLOW BASED ON THE GAUGE-UZAWA METHOD [J]. Journal of Computational Mathematics, 2022, 40(1): 26-43.
[4] Yuping Zeng, Feng Wang, Zhifeng Weng, Hanzhang Hu. A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM [J]. Journal of Computational Mathematics, 2021, 39(5): 755-776.
[5] Pouria Assari, Fatemeh Asadi-Mehregan, Mehdi Dehghan. LOCAL GAUSSIAN-COLLOCATION SCHEME TO APPROXIMATE THE SOLUTION OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS USING VOLTERRA INTEGRAL EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(2): 261-282.
[6] Mohammad Tanzil Hasan, Chuanju Xu. HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY [J]. Journal of Computational Mathematics, 2020, 38(4): 580-605.
[7] Christiane Helzel, Maximilian Schneiders. NUMERICAL APPROXIMATION OF THE SMOLUCHOWSKI EQUATION USING RADIAL BASIS FUNCTIONS [J]. Journal of Computational Mathematics, 2020, 38(1): 176-194.
[8] Jun Hu, Shangyou Zhang. AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(1): 195-222.
[9] Yubo Yang, Heping Ma. A LINEAR IMPLICIT L1LEGENDRE GALERKIN CHEBYSHEV COLLOCATION METHOD FOR GENERALIZED TIME-AND SPACE-FRACTIONAL BURGERS EQUATION [J]. Journal of Computational Mathematics, 2019, 37(5): 629-644.
[10] Tingchun Wang, Xiaofei Zhao, Mao Peng, Peng Wang. EFFICIENT AND ACCURATE NUMERICAL METHODS FOR LONG-WAVE SHORT-WAVE INTERACTION EQUATIONS IN THE SEMICLASSICAL LIMIT REGIME [J]. Journal of Computational Mathematics, 2019, 37(5): 645-665.
[11] Yuze Zhang, Yushun Wang, Yanhong Yang. THE STRUCTURE-PRESERVING METHODS FOR THE DEGASPERIS-PROCESI EQUATION [J]. Journal of Computational Mathematics, 2019, 37(4): 475-487.
[12] Ying Yang, Benzhuo Lu, Yan Xie. A DECOUPLING TWO-GRID METHOD FOR THE STEADY-STATE POISSON-NERNST-PLANCK EQUATIONS [J]. Journal of Computational Mathematics, 2019, 37(4): 556-578.
[13] Randolph E. Bank, Maximilian S. Metti. A DIAGONALLY-IMPLICIT TIME INTEGRATION SCHEME FOR SPACE-TIME MOVING FINITE ELEMENTS [J]. Journal of Computational Mathematics, 2019, 37(3): 360-383.
[14] Tiao Lu, Zhangpeng Sun. SINGULARITY-FREE NUMERICAL SCHEME FOR THE STATIONARY WIGNER EQUATION [J]. Journal of Computational Mathematics, 2019, 37(2): 170-183.
[15] Zhoufeng Wang, Peiqi Huang. AN ADAPTIVE FINITE ELEMENT METHOD FOR THE WAVE SCATTERING BY A PERIODIC CHIRAL STRUCTURE [J]. Journal of Computational Mathematics, 2018, 36(6): 845-865.
Viewed
Full text


Abstract

Journal of Computational Mathematics
Contact Us
Editor-in-Chief: Zhiming Chen
Electronic: 1991-7139
ISSN Print: 0254-9409
Issues: 6 / year
Institute of Computational Mathematics and Scientific/ Engineering Computing
Chinese Academy of Sciences
Zhongguancun East Rd. No.55
Beijing 100190, China
Phone: 86-10-82541418
Email: jcm@lsec.cc.ac.cn
×

扫码分享

京公网安备11010802040203号      京ICP备05002806号-10　　The journal is sponsored by ICMSEC　　Copyrights © ICMSEC