A Hybrid SBP-SAT/Fourier Pseudo-spectral Method for the Transient Wigner Equation Involving Inflow Boundary Conditions

In this paper, a hybrid SBP-SAT/pseudo-spectral method is proposed for solving the time-dependent Wigner equation. High-order summation-by-parts (SBP) operators are utilized to discretize the Wigner equation spatially, where the inflow boundary conditions are weakly imposed by adding simultaneous approximation terms (SATs) to the semi-discretized Wigner equation. The pseudo-differential term, governing the quantum effect, is discretized in a pseudo-spectral manner with spectral accuracy. \(L^2\)-stabilities of both the semi-discretized (excluding time) and fully discretized systems are thoroughly discussed, with the inclusion of an arbitrary-stage explicit Runge–Kutta scheme for time integration. Numerical experiments are conducted, including simulations of a harmonic oscillator, a Gaussian wave packet, and a typical RTD with its I–V characteristic curves. The numerical results demonstrate: (1) the accuracy order of the numerical scheme in discretizing the Wigner equation in phase space matches the theoretical value; (2) observation of typical quantum effects, including tunneling and negative resistance; and (3) rapid convergence of numerical solutions relative to the accuracy order of SBP operators.