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, with the Gaussian function serving as the weight function. The main challenge in this numerical scheme lies in efficiently expanding the quantum Maxwellian with the Hermite basis functions. To overcome this, we simplify the problem to the calculation 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 property and remarkable efficiency of this method.

翻译：本文提出了一种利用Hermite谱方法求解量子BGK方程的渐近保持（AP）隐式-显式（IMEX）格式。分布函数展开为Hermite多项式级数，并采用高斯函数作为权重函数。该数值格式的主要挑战在于利用Hermite基函数高效展开量子麦克斯韦分布。为克服这一难题，我们将问题简化为多对数函数的计算，并提出了一种利用Gauss-Hermite求积的高效算法。包括空间二维顶盖驱动腔流在内的多个数值模拟，验证了该方法的AP性质及显著的计算效率。