An asymptotic preserving scheme for the quantum Liouville-BGK equation

We are interested in this work in the numerical resolution of the Quantum Liouville-BGK equation, which arises in the derivation of quantum hydrodynamical models from first principles. Such models are often obtained in some asymptotic limits, for instance a diffusion or a fluid limit, and as a consequence the original Liouville equation contains small parameters. A standard method such as a split-step algorithm is then accurate provided the time step is sufficiently small compared to the asymptotic parameter, which is a severe limitation. In the case of the diffusion limit, we propose a numerical method that is accurate for time steps independent of the small parameter, and which captures well both the microscopic dynamics and the diffusion limit. Our approach is substantiated by an informal theoretical error analysis.