A Hybrid Explicit-Implicit Scheme for the Time-Dependent Wigner Equation (Journal of Computational Mathematics, 2021)
This paper designs a hybrid scheme based on finite difference methods and a spectralmethod for the time-dependent Wigner equation, and gives the error analysis for the fulldiscretization of its initial value problem. An explicit-implicit time-splitting scheme is usedfor 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 discretizationare analyzed. A Fourier spectral method is used to approximate the pseudo-differentialoperator term and the corresponding error is studied in detail. The final convergence resultshows 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 potentialbarrier are investigated. The evolution of the density function shows that a larger portionof 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.