Filbet, Francis; Hu, Jingwei; Jin, Shi A numerical scheme for the quantum Boltzmann equation with stiff collision terms. (English) Zbl 1277.82046 ESAIM, Math. Model. Numer. Anal. 46, No. 2, 443-463 (2012). The authors propose a new scheme for the quantum Boltzmann equation \[ \frac{\partial f}{\partial t} +v\cdot\nabla_xf=\frac{1}{\epsilon}Q(f),\;\;x\in\Omega\subset\mathbb{R}^{d_x},\;v\in \mathbb{R}^{d_v} . \] Here \(\epsilon\) is the Knudsen number which measures the degree of rarefication of the gas. The quantum collision operator \(Q_q\) is defined as \[ Q_q(f)(v)=\int\limits_{\mathbb{R}^{d_v}}\int\limits_{\scriptstyle{S^{d_v-1}}} B(v-v_*,\omega) [f'f'_*(1\pm\theta_0f)(1\pm\theta_0f_*)-ff_*(1\pm\theta_0f')(1\pm\theta_0f'_*)]d\omega dv_*, \] where \(\theta_0=\hbar^{d_v}\), \(\hbar\) is the rescaled Planck constant, \(\omega\) is the unit vector along \(v'-v'_*\); the collision kernel \(B\) is a nonnegative function that only depends on \(|v-v_*|\) and \(\cos\theta\).The authors idea is based on the observation that the classical Maxwellian, with the temperature replaced by the (quantum) internal energy, has the same first five moments as the quantum Maxwellian. Reviewer: Dazmir Shulaia (Tbilisi) Cited in 1 ReviewCited in 23 Documents MSC: 82C40 Kinetic theory of gases in time-dependent statistical mechanics 35Q20 Boltzmann equations 76Y05 Quantum hydrodynamics and relativistic hydrodynamics Keywords:quantum Boltzmann equation; Bose/Fermi gas; asymptotic-preserving schemes; fluid dynamic limit Citations:Zbl 1202.82066 PDFBibTeX XMLCite \textit{F. Filbet} et al., ESAIM, Math. Model. Numer. Anal. 46, No. 2, 443--463 (2012; Zbl 1277.82046) Full Text: DOI arXiv