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Volume 39, Issue 1
A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation

Haiyan Jiang, Tiao Lu & Xu Yin

DOI: 10.4208/jcm.1906-m2018-0081

J. Comp. Math., 39 (2021), pp. 22-42.

Published online: 2020-09

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  • Abstract

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.

  • Keywords

Finite difference method, Spectral method, Hybrid scheme, Error analysis, Wigner equation.

  • AMS Subject Headings

65M06, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hyjiang@bit.edu.cn (Haiyan Jiang)

tlu@math.pku.edu.cn (Tiao Lu)

2120161343@bit.edu.cn (Xu Yin)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-22, author = {Jiang , HaiyanLu , Tiao and Yin , Xu}, title = {A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {39}, number = {1}, pages = {22--42}, abstract = {

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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1906-m2018-0081}, url = {http://global-sci.org/intro/article_detail/jcm/18276.html} }
TY - JOUR T1 - A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation AU - Jiang , Haiyan AU - Lu , Tiao AU - Yin , Xu JO - Journal of Computational Mathematics VL - 1 SP - 22 EP - 42 PY - 2020 DA - 2020/09 SN - 39 DO - http://doi.org/10.4208/jcm.1906-m2018-0081 UR - https://global-sci.org/intro/article_detail/jcm/18276.html KW - Finite difference method, Spectral method, Hybrid scheme, Error analysis, Wigner equation. AB -

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.

Jiang , HaiyanLu , Tiao and Yin , Xu. (2020). A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation. Journal of Computational Mathematics. 39 (1). 22-42. doi:10.4208/jcm.1906-m2018-0081
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