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

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 benefits of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides a useful tool for solving the von Neumann equation in all regimes, near classical or not.