A Highly Efficient CN-Like Scheme for the Transient Wigner Equation

The transient Wigner equation (TWE) plays an important role in modeling quantum effects of nano-scale semiconductor devices. The pseudo-differential term contained within the TWE provides quantum descriptions for the model, however, it is costly in numerical simulations. We develop a CN-like scheme for time integration of the TWE, based on the two-step method. Additionally, the spatial discretization is the hybrid finite-difference/sinc-Galerkin scheme [H. Jiang, T. Lu, W. Yao and W. Zhang, SIAM J. Sci. Comput. 45 (2023)]. Rigorous proofs are provided to show that the CN-like scheme is unconditionally $L^2$-stable and has second-order accuracy in time. More importantly, the computational efficiency of the new CN-like scheme could be considered to be higher than that of any explicit multi-stage one-step time integration scheme. Numerical experiments are also carried out to verify the accuracy, stability and efficiency of the new CN-like scheme. In addition to the verification experiments, resonant tunneling diodes (RTDs) with various sets of parameters are simulated. I-V characteristics and associated curves are obtained to demonstrate the resonant tunneling effect and the variations in I-V characteristic curves in relation to changes in structure parameters.