Discrete-Velocity-Direction Models of BGK-type with Minimum Entropy: I. Basic Idea

Abstract

In this series of works, we develop a discrete-velocity-direction model (DVDM) with collisions of BGK-type for simulating rarefied flows. Unlike the conventional kinetic models (both BGK and discrete-velocity models), the new model restricts the transport to finite fixed directions but leaves the transport speed to be a 1-D continuous variable. Analogous to the BGK equation, the discrete equilibriums of the model are determined by minimizing a discrete entropy. In this first paper, we introduce the DVDM and investigate its basic properties, including the existence of the discrete equilibriums and the H-theorem. We also show that the discrete equilibriums can be efficiently obtained by solving a convex optimization problem. The proposed model provides a new way in choosing discrete velocities for the computational practice of the conventional discrete-velocity methodology. It also facilitates a convenient multidimensional extension of the extended quadrature method of moments. We validate the model with numerical experiments for two benchmark problems at moderate computational costs.

Realizabily of the DVDM

In this appendix, we show that for the two-dimensional DVDM, more macroscopic states can be realized with more discrete-velocity directions. In this case, the discrete-velocity directions can be expressed as $\varvec{l}_m = (\cos \gamma _m, \sin \gamma _m)^T$ with $\gamma _m \in [0,\pi )$ and therefore can be understood as the complex numbers $e^{i \gamma _m}$. Without loss of generality, we assume $0 \le \gamma _1< \dots< \gamma _N < \pi$. The main result here is the following lemma.

Lemma 4

For any N-tuple $(a_m)_{m=1}^N \in {\mathbb {R}}$ such that

$$\begin{aligned} \sum _{m=1}^N a_m e^{i \gamma _m} = 1, \end{aligned}$$

we have

$$\begin{aligned} \sum _{m=1}^N |a_m| \ge \frac{\sin \gamma _1 + \sin \gamma _N}{\sin (\gamma _N - \gamma _1)}. \end{aligned}$$

(43)

The minimum is attained when

$$\begin{aligned} a_1=\frac{\sin \gamma _N}{\sin (\gamma _N - \gamma _1)}, \ a_N=\frac{-\sin \gamma _1}{\sin (\gamma _N - \gamma _1)}, \text { and } a_2=\dots =a_{N-1}=0. \end{aligned}$$

Proof

Set $z_m = a_m e^{i \gamma _m}$. According to the assumption, we have $\sum _m z_m = 1$ and

$$\begin{aligned} \sum _m |a_m| = \sum _m |z_m| \ge \left|\sum _m z_m \right|= 1. \end{aligned}$$

Therefore the lemma obviously holds with $\gamma _1=0$.

For $\gamma _1>0$, since $\gamma _1 \le \gamma _m<\pi$ for all m, $z_m$ is not a real number unless $z_m=0$. Thus, from $\sum _m z_m=1$ it follows that there must be some $z_m$ with positive imaginary part (denote their sum by $z^+$) and some $z_m$ with negative imaginary part (denote their sum by $z^-$). Obviously, we have $\gamma _+:= \arg z^+ \in (0,\pi )$ and $\gamma _-:= \arg z^- \in (-\pi ,0)$. Note that $z^+ + z^- = 1$, meaning that the three complex numbers $z^+$, $z^-$ and 1 constitute a triangle (after a proper shift of $z^-$) in the complex plane. Then the angle $(\gamma _+ - \gamma _-)$ from $z^-$ to $z^+$ is in $(0, \pi )$. We thus see from the law of sines:

$$\begin{aligned} \frac{1}{\sin (\gamma _+ - \gamma _-)} = \frac{|z^+|}{\sin (- \gamma _-)} = \frac{|z^-|}{\sin \gamma _+} \end{aligned}$$

that

$$\begin{aligned} \sum _m |a_m| = \sum _m |z_m| \ge |z^+| + |z^-| = \frac{\sin \gamma _+ - \sin \gamma _-}{\sin (\gamma _+ - \gamma _-)} =: S(\gamma _+, \gamma _-), \end{aligned}$$

which is monotonically increasing with $\gamma _+$ and decreasing with $\gamma _-$. This can be seen by computing the derivatives:

$$\begin{aligned} \begin{aligned} \frac{\partial S}{\partial \gamma _+}&= \frac{\cos \gamma _+ \sin (\gamma _+ - \gamma _-) - (\sin \gamma _+ - \sin \gamma _-) \cos (\gamma _+ - \gamma _-)}{\sin ^2(\gamma _+ - \gamma _-)} \\ {}&= \frac{\sin \gamma _- (\cos (\gamma _+ - \gamma _-) - 1)}{\sin ^2(\gamma _+ - \gamma _-)} \ge 0, \end{aligned} \end{aligned}$$

and similarly $\frac{\partial S}{\partial \gamma _-} \le 0$. Geometrically we can easily see that $\gamma _+ \ge \gamma _1$ and $\gamma _- \le \gamma _N - \pi$. Thus, the minimum of $|z^+| + |z^-|$ is attained when $\gamma _+ = \gamma _1$ and $\gamma _- = \gamma _N - \pi$. This corresponds to $a_2=\dots =a_{N-1}=0$ and leads exactly to the RHS of Eq. (43). $\square$

With this lemma, we can explicitly write down the constraint on temperature $\theta$. To do this, we denote by $\gamma _{\varvec{U}} \in (-\pi ,\pi ]$ the argument of the flow velocity $\varvec{U}$ in Eq. (17). Then the second equality in Eq. (17) can be rewritten as

$$\begin{aligned} \sum _{m=1}^N {\hat{\rho _m}} {{\hat{u}}}_m e^{i \gamma _m} = \rho U e^{i \gamma _{\varvec{U}}} \end{aligned}$$

or

$$\begin{aligned} \sum _{m=1}^N s{\hat{\rho _m}} {{\hat{u}}}_m e^{i (\gamma _m - \eta )} = \rho U, \end{aligned}$$

(44)

where

$$\begin{aligned} s = \left\{ \begin{aligned} 1, \quad&\text {if } 0< \gamma _{\varvec{U}} \le \pi , \\ -1, \quad&\text {if } -\pi < \gamma _{\varvec{U}} \le 0, \end{aligned} \right. \end{aligned}$$

and $\eta = \gamma _{\varvec{U}} - \frac{s-1}{2}\pi \in (0,\pi ]$.

Let k ($1\le k \le N$) be such that

$$\begin{aligned} \gamma _k < \eta \le \gamma _{k+1}. \end{aligned}$$

For convenience, we define $\gamma _{N+1}=\pi + \gamma _1$. Then we set

$$\begin{aligned} {{\tilde{\gamma }}}_m = \left\{ \begin{aligned}&\gamma _m + \pi - \eta ,&1\le m\le k, \\ {}&\gamma _m - \eta ,&k+1\le m \le N. \end{aligned} \right. \end{aligned}$$

It is clear that $0\le {{\tilde{\gamma }}}_{k+1}<\cdots< {{\tilde{\gamma }}}_N< {{\tilde{\gamma }}}_1< \cdots< {{\tilde{\gamma }}}_k < \pi$. Now we apply the lemma above to Eq. (44) to get

$$\begin{aligned} U^2 \left( \frac{\sin {{\tilde{\gamma }}}_{k+1} + \sin \tilde{\gamma }_k}{\sin ({{\tilde{\gamma }}}_k - {{\tilde{\gamma }}}_{k+1})} \right) ^2 \le \frac{1}{\rho ^2}\left( \sum _{m=1}^N |{\hat{\rho _m}} {{\hat{u}}}_m| \right) ^2 < 2 E, \end{aligned}$$

where the last inequality is Eq. (18). The left-hand side is

$$\begin{aligned} U^2\left( \frac{\sin (\gamma _{k+1}-\eta ) + \sin (\eta - \gamma _k)}{\sin (\gamma _{k+1} - \gamma _k)} \right) ^2, \end{aligned}$$

which obviously goes to $U^2$ as $\gamma _{k+1}-\gamma _k \rightarrow 0$. Moreover, the constraint for $\theta = (2E - U^2)/2$ becomes

$$\begin{aligned} \theta > U^2 \frac{\sin (\gamma _{k+1}-\eta ) \sin (\eta - \gamma _k)}{1+ \cos (\gamma _{k+1} - \gamma _k)}. \end{aligned}$$

(45)

This lower bound approaches 0 when $\gamma _{k+1} - \gamma _k \rightarrow 0$.