Singularity-Free Numerical Scheme for the Stationary Wigner Equation (Journal of Computational Mathematics, 2018)
For the stationary Wigner equation with inflow boundary conditions, the numericalconvergence with respect to the velocity mesh size are deteriorated due to the singularityat velocity zero. In this paper, using the fact that the solution of the stationary Wignerequation is subject to an integral constraint, we prove that the Wigner equation can bewritten 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 discreteoperators discretizing $\mathcal{B}[V]$ are proved to be uniformly bounded with respect to the meshsize. Based on the theoretical findings, a singularity-free numerical method is proposed.Numerical results are provided to show our improved numerical scheme performs muchbetter in numerical convergence than the original scheme based on discretizing $\mathcal{A}[V]$.
Keywords: stationary Wigner equation; singularity-free; numerical convergence.