On the approximation of the von Neumann equation in the semiclassical limit. Part II : numerical analysis
This work offers a numerical solution for quantum systems in the semiclassical regime, though it is incremental as it builds on a previously proposed method.
The paper analyzes a Hermite spectral method for approximating the von Neumann equation in the semiclassical limit, addressing stiffness through Weyl's variables and providing error estimates based on regularity propagation.
This paper is devoted to the numerical analysis of the Hermite spectral method proposed in [14], which provides, in the semiclassical limit, an asymptotic preserving approximation of the von Neumann equation. More precisely, it relies on the use of so-called Weyl's variables to effectively address the stiffness associated to the equation. Then by employing a truncated Hermite expansion of the density operator, we successfully manage this stiffness and provide error estimates by leveraging the propagation of regularity in the exact solution.