On the approximation of the von Neumann equation in the semi-classical limit. Part I : numerical algorithm

TL;DR

This paper introduces a novel numerical method for discretizing the von Neumann equation that remains efficient across semi-classical and classical regimes, utilizing Weyl's variables, Hermite expansions, and finite volume techniques.

Contribution

The paper presents an asymptotic preserving numerical scheme combining Weyl's variables and Hermite expansions for the von Neumann equation, improving efficiency in the semi-classical limit.

Findings

The method effectively handles stiffness in the von Neumann equation.

Numerical simulations demonstrate the approach's efficiency across regimes.

The scheme is practical for implementation with finite volume approximation.

Abstract

We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl's variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation for practical implementation and conduct numerical simulations to illustrate the efficiency of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides an efficient tool for solving the von Neumann equation in all regimes, near classical or not.