Singularity-free Numerical Scheme for the Stationary Wigner Equation

TL;DR

This paper introduces a new numerical scheme for the stationary Wigner equation that avoids singularities at zero velocity, improving convergence and stability compared to traditional methods.

Contribution

The authors reformulate the stationary Wigner equation into a bounded operator form, enabling a singularity-free numerical scheme with proven uniform boundedness and better convergence.

Findings

The new scheme demonstrates significantly improved numerical convergence.

The reformulation ensures the operator remains bounded regardless of mesh size.

Numerical results confirm the scheme's superior performance over traditional methods.

Abstract

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 $\mathcal{A}[V]=\Theta[V]/v$ 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 $\mathcal{A}[V]$.