An explicit multistep method for the Wigner problem

TL;DR

This paper introduces an explicit multistep method for solving the Wigner problem that leverages spectral collocation and FFTs, enabling efficient and accurate simulations without CFL restrictions.

Contribution

The paper presents a novel explicit multistep scheme combining spectral collocation and FFTs for the Wigner problem, improving computational efficiency and accuracy.

Findings

The method achieves high accuracy in numerical examples.

It reduces computational complexity via FFT-based potential calculations.

Time stepping is unrestricted by CFL conditions.

Abstract

An explicit multistep scheme is proposed for solving the initial-value Wigner problem. In this scheme, the integrated form of the Wigner equation is approximated by extrapolation or interpolation polynomials on backwards characteristics, and the pseudo-differential operator is tackled by the spectral collocation method. Since it exploits the exact Lagrangian advection, the time stepping of the multistep scheme is not restricted by the CFL-type condition. It is also demonstrated that the calculations of the Wigner potential can be carried out by two successive FFTs, thereby reducing the computational complexity dramatically. Numerical examples illustrating its accuracy are presented.