Numerical schemes for a multi-species quantum BGK model

This work is devoted to the numerical implementation of the quantum Bhatnagar- Gross-Krook (BGK) model for gas mixtures consisting of classical and quantum particles (fermions, bosons). We consider the model proposed by Bae, Klingenberg, Pirner, and Yun in 2021 and implement an Implicit-Explicit (IMEX) scheme due to the stiffness of the collision operator. A major obstacle is updating the parameters of quantum local equilibrium, which requires computing by inverting the relation between density and energy at every grid point in space and time. We address this difficulty by using the Lagrange multiplier method to minimize a potential function subject to constraints defined by specific moment equalities. Moreover, we analyze the convergence of mean velocity and temperature between the species both analytically and numerically. When a quantum component is included, we observe that the converging quantity is physical temperature, not the kinetic temperature. This differs from the mixture of classical species.