A numerical scheme for the quantum Boltzmann equation with stiff collision terms (Q2846152)

The authors propose a new scheme for the quantum Boltzmann equation NEWLINE\[NEWLINE\frac{\partial f}{\partial t} +v\cdot\nabla_xf=\frac{1}{\epsilon}Q(f),\;\;x\in\Omega\subset\mathbb{R}^{d_x},\;v\in \mathbb{R}^{d_v} .NEWLINE\]NEWLINE Here \(\epsilon\) is the Knudsen number which measures the degree of rarefication of the gas. The quantum collision operator \(Q_q\) is defined as NEWLINE\[NEWLINEQ_q(f)(v)=\int\limits_{\mathbb{R}^{d_v}}\int\limits_{\scriptstyle{S^{d_v-1}}} B(v-v_*,\omega) [f'f'_*(1\pm\theta_0f)(1\pm\theta_0f_*)-ff_*(1\pm\theta_0f')(1\pm\theta_0f'_*)]d\omega dv_*, NEWLINE\]NEWLINE 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 \(\cos\theta\).NEWLINENEWLINEThe 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.