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.

翻译：我们提出了一种离散化冯·诺依曼方程的新方法，该方法在半经典极限下是高效的。此方法首先基于所谓的魏尔变量来处理方程固有的刚性。随后，通过对密度算子进行截断的埃尔米特展开，我们成功克服了这一刚性。此外，我们开发了一种用于实际实现的有限体积近似方法，并进行了数值模拟以展示我们方法的效率。这种渐近保持的数值近似，结合埃尔米特多项式的使用，为求解冯·诺依曼方程提供了一个高效的工具，无论其处于近经典区域还是其他区域。