Singularity-free Numerical Scheme for the Stationary Wigner Equation
This work addresses a numerical bottleneck in simulating quantum transport via the Wigner equation, offering a practical improvement for computational physicists and engineers.
The stationary Wigner equation suffers from numerical convergence issues due to a singularity at velocity zero. The authors reformulated the equation using a bounded operator and proposed a singularity-free numerical scheme that achieves significantly better convergence than the original approach.
For the stationary Wigner equation with inflow boundary conditions, its numerical convergence with respect to the velocity mesh size are deteriorated due to the singularity at velocity zero. In this paper, using the fact that the solution of the stationary Wigner equation is subject to an algebraic constraint, we prove that the Wigner equation can be written into a form with a bounded operator $\mathcal{B}[V]$, which is equivalent to the operator $\mathcal{A}[V]=Θ[V]/v$ in the original Wigner equation under some conditions. Then the discrete operators discretizing $\mathcal{B}[V]$ are proved to be uniformly bounded with respect to the mesh size. Based on the therectical findings, a signularity-free numerical method is proposed. Numerical reuslts are proivded to show our improved numerical scheme performs much better in numerical convergence than the original scheme based on discretizing $\mathcal{A}[V]$.