Asymptotic-Preserving Exponential Methods for the Quantum Boltzmann Equation with High-Order Accuracy

Abstract

In this paper we develop high order asymptotic preserving methods for the spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li and Pareschi (J Comput Phys 259:402–420, 2014) where asymptotic preserving exponential Runge–Kutta methods for the classical inhomogeneous Boltzmann equation were constructed. A major difficulty here is related to the non Gaussian steady states characterizing the quantum kinetic behavior. We show that the proposed schemes achieve high-order accuracy uniformly in time for all Planck constants ranging from classical regime to quantum regime, and all Knudsen number ranging from kinetic regime to fluid regime. Computational results are presented for both Bose gas and Fermi gas.

Appendix: Derivation of \(\partial _t \mathcal {M}_q\)

In this appendix, we give the details of the derivation of (3.8). Our goal is to represent $\partial _tz$ and $\partial _tT$ in Eq. (3.7) in terms of $\partial _t\rho$ and $\partial _te$.

First, combining the two equations in system (2.14) gives

$$\begin{aligned} \frac{Q_{\frac{d}{2}}^{\frac{d}{2}+1}(z)}{Q_{\frac{d}{2}+1}^{\frac{d}{2}}(z)}=\theta _0 \left( \frac{d}{4\pi e}\right) ^{\frac{d}{2}} \rho . \end{aligned}$$

(7.1)

Therefore, we define a function $F(z)$ such that

$$\begin{aligned} y=F(z)=\frac{Q_{\frac{d}{2}}^{\frac{d}{2}+1}(z)}{Q_{\frac{d}{2}+1}^{\frac{d}{2}}(z)}, \end{aligned}$$

(7.2)

and a function $G(y)$ such that

$$\begin{aligned} z=G(y)=F^{-1}(y). \end{aligned}$$

(7.3)

Then we have

$$\begin{aligned} G'(y)=\frac{1}{F'(z)}=\frac{Q_{\frac{d}{2}+1}^d(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}}^{\frac{d}{2}}(z)Q_{\frac{d}{2}}'(z)Q_{\frac{d}{2}+1}^{\frac{d}{2}}(z)-\frac{d}{2}Q_{\frac{d}{2}+1}^{\frac{d}{2}-1}(z)Q_{\frac{d}{2}+1}'(z)Q_{\frac{d}{2}}^{\frac{d}{2}+1}(z)}. \end{aligned}$$

(7.4)

For the Bose–Einstein/Fermi–Dirac function, one has the following nice property (see [30])

$$\begin{aligned} zQ_{\nu }'(z)=Q_{\nu -1}(z). \end{aligned}$$

(7.5)

Using (7.5) in (7.4),

$$\begin{aligned} G'(y)&=\frac{1}{F'(z)}=\frac{zQ_{\frac{d}{2}+1}^d(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}}^{\frac{d}{2}}(z)Q_{\frac{d}{2}-1}(z)Q_{\frac{d}{2}+1}^{\frac{d}{2}}(z)-\frac{d}{2}Q_{\frac{d}{2}+1}^{\frac{d}{2}-1}(z)Q_{\frac{d}{2}}(z)Q_{\frac{d}{2}}^{\frac{d}{2}+1}(z)} \nonumber \\&=\frac{zQ_{\frac{d}{2}+1}^{\frac{d}{2}+1}(z)}{Q_{\frac{d}{2}}^{\frac{d}{2}}(z)\left[ \left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)Q_{\frac{d}{2}+1}(z)-\frac{d}{2}Q_{\frac{d}{2}}^{2}(z)\right] }. \end{aligned}$$

(7.6)

From the second equation of (2.14) we know

$$\begin{aligned} Q_{\frac{d}{2}+1}(z)=\frac{2e}{dT}Q_{\frac{d}{2}}(z), \end{aligned}$$

(7.7)

then

$$\begin{aligned} G'(y)=\frac{z\left( \frac{2e}{dT}\right) ^{\frac{d}{2}}}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)}. \end{aligned}$$

(7.8)

Note that

$$\begin{aligned} z=G\left( \theta _0 \left( \frac{d}{4\pi e}\right) ^{\frac{d}{2}} \rho \right) , \quad T=\frac{\theta _0^{\frac{2}{d}}}{2\pi }\left( \frac{\rho }{Q_{\frac{d}{2}}(z)}\right) ^{\frac{2}{d}}, \end{aligned}$$

(7.9)

so we have

$$\begin{aligned} \partial _t z&=G'\left( \theta _0 \left( \frac{d}{4\pi e}\right) ^{\frac{d}{2}} \rho \right) \theta _0\left( \frac{d}{4\pi }\right) ^{\frac{d}{2}}\left( \frac{1}{e^{\frac{d}{2}}}\partial _t\rho -\frac{d}{2}\frac{\rho }{e^{\frac{d}{2}+1}}\partial _te\right) \nonumber \\&=\frac{zQ_{\frac{d}{2}}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)}\left( \frac{1}{\rho }\partial _t\rho -\frac{d}{2e}\partial _te\right) , \end{aligned}$$

(7.10)

and

$$\begin{aligned} \partial _tT&= \frac{\theta _0^{\frac{2}{d}}}{\pi d}\left( \frac{\rho ^{\frac{2}{d}-1}}{Q_{\frac{d}{2}}^\frac{2}{d}(z)}\partial _t\rho -\frac{\rho ^{\frac{2}{d}}Q_{\frac{d}{2}}'(z)}{Q_{\frac{d}{2}}^{\frac{2}{d}+1}(z)}\partial _tz\right) =\frac{\theta _0^{\frac{2}{d}}}{\pi d}\left( \frac{\rho ^{\frac{2}{d}-1}}{Q_{\frac{d}{2}}^\frac{2}{d}(z)}\partial _t\rho -\frac{\rho ^{\frac{2}{d}}Q_{\frac{d}{2}-1}(z)}{zQ_{\frac{d}{2}}^{\frac{2}{d}+1}(z)}\partial _tz\right) \nonumber \\&= \frac{2T}{d}\frac{1}{\rho }\rho _t -\frac{2T}{d}\frac{Q_{\frac{d}{2}-1}(z)}{Q_{\frac{d}{2}}(z)}\frac{1}{z}\partial _tz=\frac{2T}{d}\frac{1}{\rho }\rho _t \nonumber \\&-\frac{2T}{d}\frac{Q_{\frac{d}{2}-1}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)}\left( \frac{1}{\rho }\partial _t\rho -\frac{d}{2e}\partial _te\right) . \end{aligned}$$

(7.11)

Therefore,

$$\begin{aligned}&\frac{1}{z}\partial _tz+\frac{(v-u)^2}{2T^2}\partial _tT=\frac{Q_{\frac{d}{2}}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)}\left( \frac{1}{\rho }\partial _t\rho -\frac{d}{2e}e_t\right) \nonumber \\&+\frac{(v-u)^2}{dT}\frac{1}{\rho }\partial _t\rho -\frac{(v-u)^2}{dT}\frac{Q_{\frac{d}{2}-1}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)}\left( \frac{1}{\rho }\partial _t\rho -\frac{d}{2e}\partial _te\right) \nonumber \\&=\left[ \frac{Q_{\frac{d}{2}}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)}+ \frac{(v-u)^2}{dT}\left( 1-\frac{Q_{\frac{d}{2}-1}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)} \right) \right] \frac{1}{\rho }\partial _t\rho \nonumber \\&+\left[ \frac{(v-u)^2}{2eT}\frac{Q_{\frac{d}{2}-1}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)} -\frac{d}{2e}\frac{Q_{\frac{d}{2}}(z)}{\left( \frac{d}{2}+1\right) Q_{\frac{d}{2}-1}(z)-\frac{d^2T}{4e}Q_{\frac{d}{2}}(z)} \right] \partial _te. \end{aligned}$$

(7.12)

Then if we define $M(z)$ and $N(z)$ as in (3.12), (3.8) follows readily from the above equation.