A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation
A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation
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DOI：
10.4208/jcm.1906-m2018-0081
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发表时间：
2021-06
影响因子：
0.9
通讯作者：
Haiyan Jiang
中科院分区：
文献类型：
--
作者：
Haiyan Jiang
本文设计了一种基于有限差分法和谱法的时变Wigner方程的混合格式，并对其初值问题进行了完全离散化的误差分析。时间积分采用显式-隐式时间分裂格式，平流项离散采用二阶迎风有限差分格式。分析了全离散化的一致性误差和稳定性。采用傅里叶谱法对伪微分算子项进行近似，并详细研究了相应的误差。最后的收敛结果清楚地显示了解的正则性如何影响所提方案的收敛顺序。数值结果证实了分析的清晰性。研究了高斯波包穿过高斯势垒时的散射效应。密度函数的演化表明，当屏障的高度和宽度增加时，反射的波的比例增大。数学学科分类：65M06、65M70。
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. Mathematics subject classification: 65M06, 65M70.