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This paper presents an asymptotic preserving (AP) implicit-explicit (IMEX)scheme for solving the quantum BGK equation using the Hermite spectral method.The distribution function is expanded in a series of Hermite polynomials, withthe Gaussian function serving as the weight function. The main challenge inthis numerical scheme lies in efficiently expanding the quantum Maxwellian withthe Hermite basis functions. To overcome this, we simplify the problem to thecalculation of polylogarithms and propose an efficient algorithm to handle it,utilizing the Gauss-Hermite quadrature. Several numerical simulations,including a spatially 2D lid-driven cavity flow, demonstrate the AP propertyand remarkable efficiency of this method.

A highly efficient asymptotic preserving IMEX method for the quantum BGK equation