这是自己的一份笔记，主要是和格子Boltzmann方法的DnQb离散速度模型相关的。还请读者确定自己已经知道常见的DnQb模型长什么样再来读这篇文章，这样会好理解一些。

Gauss-Hermite积分与DnQb模型

一般来说，密度分布函数的Maxwell-Boltzmann平衡态分布为：

$$f^{eq} = \frac{\rho}{ (2 \pi R T)^{D/2} } \exp \left[ - \frac{(\boldsymbol{\xi} - \boldsymbol{u})^2}{2 R T} \right], \tag{1-1}$$

其中 $\boldsymbol{\xi}$ 为分子速度；$R$ 为气体常数；$D$为问题的维度数；$\rho$ 为流场密度；$\boldsymbol{u}$ 为流场速度。而 LBM 里常用的离散平衡态分布一般写作（沿着 $\boldsymbol{c}_\alpha$ 方向，权重为 $W_\alpha$，$c_s^2 = RT$）：

$$f^{eq}_{\alpha} = \rho W_{\alpha} \times \left[ 1 + \frac{\boldsymbol{\xi}_{\alpha} \cdot \boldsymbol{u}}{c_s^2} + \frac{(\boldsymbol{\xi}_{\alpha} \cdot \boldsymbol{u})^2}{2 c_s^4} - \frac{|\boldsymbol{u}|^2}{2 c_s^2} \right]. \tag{1-2}$$

那么这两者是怎么联系起来的呢？参考文献 [1] 讲的就是这个事情。最近自己总算是略懂一点了，这里把笔记写下来，作为备忘录。

平衡态分布的展开

当马赫数远小于1时，我们可以对式(1)的平衡态做如下离散：

$$\begin{aligned} f^{eq} (\boldsymbol{\xi}) &= \frac{\rho}{ (2 \pi R T)^{D/2} } \exp \left( - \frac{|\boldsymbol{\xi}|^2}{2 R T} \right)\\ & \quad \times \left[ 1 + \frac{\boldsymbol{\xi} \cdot \boldsymbol{u}}{R T} + \frac{(\boldsymbol{\xi} \cdot \boldsymbol{u})^2}{2 (R T)^2} - \frac{|\boldsymbol{u}|^2}{2 R T} \right] \end{aligned} \tag{1-3}$$

式(1-3)中，我们对 $\boldsymbol{u}$ 进行展开，并且只截断到2阶。而格子Boltzmann方法（LBM）要做的，便是把下面的这个积分

$$\int_{-\infty}^{+\infty} \psi(\boldsymbol{\xi}) f^{eq} (\boldsymbol{\xi}) \mathrm{d}\boldsymbol{\xi}$$

用离散的方式近似计算，其中 $\psi(\boldsymbol{\xi})$ 是关于 $\boldsymbol{\xi}$ 的多项式。下面我们将先以一维的情况为例，不去考虑多维的积分，简单讲讲数值离散的具体操作。

一维情形的离散 —— 以 D1Q3 为例

由于这一节只讨论一维问题，分子速度 $\xi$ 和宏观速度 $u$ 即是标量值（不需要向量形式的加粗），并且维度 $D=1$。所以，式(3)写作

$$\begin{aligned} f^{eq} (\xi) &= \frac{\rho}{ (2 \pi R T)^{1/2} } \exp \left( - \frac{\xi^2}{2 R T} \right)\\ & \quad \times \left[ 1 + \frac{\xi u}{R T} + \frac{(\xi u)^2}{2 (R T)^2} - \frac{u^2}{2 R T} \right]. \end{aligned} \tag{1-4}$$

积分形式的抽象

式(1-4)的中括号内的四项可以用积分的加法拆成四个积分，并且每一项都可以看成如下形式：

$$\text{Constant} \cdot \int_{-\infty}^{+\infty} \exp \left( - \frac{\xi^2}{2 R T} \right) \psi(\xi) \mathrm{d}\xi,$$

其中 $\psi(\xi)$ 为关于 $\xi$ 的多项式。那么，我们令 $\psi_{m}(\xi) = \xi^{m}$ （$m$ 为自然数），将研究目标变为式(1-5)的通式，即：

$$\frac{\rho}{ (2 \pi R T)^{1/2} } \cdot \int_{-\infty}^{+\infty} \exp \left( - \frac{\xi^2}{2 R T} \right) \psi_{m}(\xi) \mathrm{d}\xi. \tag{1-5}$$

为简化这个积分的计算，这里需要引入变量代换

$$\zeta = \xi / \sqrt{2 R T},$$

则

$$\mathrm{d}\xi = \sqrt{2 R T} \mathrm{d}\zeta.$$

代换后，有：

$$\begin{aligned} & \frac{\rho}{ (2 \pi R T)^{1/2} } \cdot \int_{-\infty}^{+\infty} \exp \left( - \zeta^2 \right) \psi_{m}(\sqrt{2 R T} \zeta) \sqrt{2 R T} \mathrm{d}\boldsymbol{\zeta} \\=& \frac{\rho}{ (2 \pi R T)^{1/2} } \cdot (2 R T)^{(m+1)/2} \cdot I_m \\=& \frac{\rho}{\sqrt{\pi}} \cdot (2 R T)^{m/2} \cdot I_m \end{aligned}$$

其中 $\displaystyle I_m = \int_{-\infty}^{+\infty} \exp \left( - \zeta^2 \right) \zeta^m \mathrm{d}\boldsymbol{\zeta}$，且 $m$ 为自然数。

所以，下面的积分会被展开为

$$\begin{aligned} I=& \int_{-\infty}^{+\infty} \psi_m(\xi) f^{eq} (\xi) \mathrm{d}\xi \\=& \frac{\rho}{\sqrt{\pi}} \left[ \left(1 - \frac{u^2}{2 R T}\right) (2 R T)^{m/2} I_m + \frac{u}{RT} (2 R T)^{(m+1)/2} I_{m+1} \right. \\&\qquad \left. + \frac{u^2}{2(RT)^2} (2 R T)^{(m+2)/2} I_{m+2} \right]. \end{aligned} \tag{1-6}$$

数值积分的构造

假设我们想用一个三点格式的Gauss-Hermite数值积分进行近似，则：

$$\int_{-\infty}^{+\infty} \exp(- \zeta^2) \phi(\zeta) \mathrm{d} \zeta = \sum_{i \in [-1, 0, 1]} w_{i} \phi(\zeta_{i}).$$

其中：

$$\begin{aligned} \zeta_0 = 0 ,\quad& \zeta_{1} = \sqrt{3/2} ,\quad& \zeta_{-1} = -\sqrt{3/2} ,\\ w_0 = 2\sqrt{\pi} / 3 ,\quad& w_{1} = \sqrt{\pi} / 6 ,\quad& w_{-1} = \sqrt{\pi} / 6. \end{aligned}$$

如果用变换关系代换回 $\xi$，则 $\xi_0 = 0$, $\xi_{-1} = -\sqrt{3RT}$ 和 $\xi_{1} = \sqrt{3RT}$.

所以，积分 $I_m$ 可以被近似成：

$$\begin{aligned} I_m &= \int_{-\infty}^{+\infty} \exp \left( - \zeta^2 \right) \zeta^m \mathrm{d}\boldsymbol{\zeta} \\&\approx \sum_{i \in [-1, 0, 1]} w_{i} \zeta_{i}^m \\&= (2RT)^{-m/2} \sum_{i \in [-1, 0, 1]} w_{i} \xi_{i}^m \end{aligned}$$

也就是说，

$$(2 R T)^{m/2} I_m \approx \sum_{i \in [-1, 0, 1]} w_{i} \xi_{i}^m$$

综上所述，式(1-6)就会被改写为：

$$\begin{aligned} I=& \int_{-\infty}^{+\infty} \psi_m(\xi) f^{eq} (\xi) \mathrm{d}\xi \\ \approx& \sum_{i \in [-1, 0, 1]} \xi_{i}^{m} \cdot \left\{ \rho \frac{w_{i}}{\sqrt{\pi}} \cdot \left[ 1 + \frac{\xi_{i} u}{R T} + \frac{(\xi_{i} u)^2}{2 (R T)^2} - \frac{u^2}{2 R T} \right] \right\}. \end{aligned}$$

到这一步，我们就可以定义 $\xi_{i}$ 方向的离散平衡态 $f^{eq}_{i}$ 为：

$$f^{eq}_{i} = \rho W_{i} \cdot \left[ 1 + \frac{\xi_{i} u}{c_s^2} + \frac{(\xi_{i} u)^2}{2 c_s^4} - \frac{u^2}{2 c_s^2} \right]. \tag{1-7}$$

其中的权重 $W_{i} = w_i / \sqrt{\pi}$。并且，如果令 $c = \sqrt{3RT}$， 则 $c_s = \sqrt{RT} = c / \sqrt{3}$ 。

至此，我们就整理出了 D1Q3 模型：

${i}$	离散速度 $\xi_{i}$	权重系数 $W_i$
0	0	2/3
-c	-c	1/6
c	c	1/6

注意：在多数LBM实践中，我们通常将 $c$ 无量纲化，即令 $c=1$。

更高维的扩展

由于各个坐标轴之间是正交的，这种做法同样可以拓展到高维空间。

打个比方，对于三维问题， $\boldsymbol{\xi} = [\xi_x, \xi_y, \xi_z]^{\mathrm{T}}$， 则 $\mathrm{d}\boldsymbol{\xi} = \mathrm{d}\xi_x \mathrm{d}\xi_y \mathrm{d}\xi_z$。积分时就只需要将不同方向的分量分离出来就可以了。

对于二维问题，记 $\psi_{m,n}(\boldsymbol{\xi})=\xi_x^{m} \xi_y^{n}$，则

$$\begin{aligned} I =& \int\psi_{m , n} (\boldsymbol{\xi}) f^{( \mathrm{s t} )} \mathrm{d}\boldsymbol{\xi} \\=& \frac{\rho}{\pi} ( \sqrt{2 R T} )^{m+n} \Biggl\{\left( 1-\frac{u^{2}} {2 R T} \right) I_{m} I_{n} \\&+ \frac{1} {\sqrt{2 R T}} 2 ( u_{x} I_{m+1} I_{n}+u_{y} I_{m} I_{n+1} ) \\&+ \frac{1} {R T} (u_{x}^{2} I_{m+2} I_{n}+2 u_{x} u_{y} I_{m+1} I_{n+1}+u_{y}^{2} I_{m} I_{n+2}) \Biggr\} \end{aligned} \tag{1-8}$$

式(1-8)同样可以整理成：

$$I = \sum_{i,j\in[-1,0,1]} \psi(\boldsymbol{\xi}_{i,j}) \cdot \left\{ \rho \frac{w_i w_j}{\pi} \left[ 1 + \frac{\boldsymbol{\xi}_{i,j} \cdot \boldsymbol{u}}{c_s^2} + \frac{(\boldsymbol{\xi}_{i,j} \cdot \boldsymbol{u})^2}{2 c_s^4} - \frac{\|\boldsymbol{u}\|^2}{2 c_s^2} \right] \right\} \tag{1-9}$$

其中 $\boldsymbol{\xi}_{i,j}=\sqrt{2RT} (\zeta_i,\zeta_j)$。令 $W_{i,j} = w_i w_j / \pi$ ，便得出了 D2Q9 模型的权重。

类似的，对于三维问题，则可以整理出

$$I = \sum_{i,j,k\in[-1,0,1]} \psi(\boldsymbol{\xi}_{i,j,k}) \cdot \left\{ \rho \frac{w_i w_j w_k}{\pi^{3/2}} \left[ 1 + \frac{\boldsymbol{\xi}_{i,j,k} \cdot \boldsymbol{u}}{c_s^2} + \frac{(\boldsymbol{\xi}_{i,j,k} \cdot \boldsymbol{u})^2}{2 c_s^4} - \frac{\|\boldsymbol{u}\|^2}{2 c_s^2} \right] \right\} \tag{1-10}$$

其中 $\boldsymbol{\xi}_{i,j,k}=\sqrt{2RT} (\zeta_i,\zeta_j,\zeta_k)$。令 $W_{i,j,k} = w_i w_j w_k / \pi^{3/2}$ ，便得出了 D3Q27 模型的权重。

待定系数法与DnQb模型

上面的内容可以说从数学上给出了明确的证明。然而实际使用中，其实还是用待定系数法来敲定 DnQb 模型更多一些。下面的内容里，为了保持和何雅玲老师的书标号一致，这里将第 $\alpha$ 个格子速度向量记作 $\boldsymbol{e}_{\alpha}$，其权重也记作 $\omega_{\alpha}$。

一般来说，d 维空间下的格子速度的 n 阶张量为：

$$E_{i_1 i_2 ... i_n}^{(n)} = \sum_{\alpha} (\boldsymbol{e}_{\alpha})_{i_1} (\boldsymbol{e}_{\alpha})_{i_2} ... (\boldsymbol{e}_{\alpha})_{i_n} \tag{2-1}$$

假如你选取的离散速度集（共有 $M$ 个速度）是各向同性的，则应该满足

$$E^{(2n+1)}_{i_1 i_2 ... i_{2n+1}} = 0,$$

和

$$E^{(2n)}_{i_1 i_2 ... i_{2n}} = \frac{M}{d (d+2) ... (d+2n-2)} \Delta^{(2n)}_{i_1 i_2 ... i_{2n}}$$

下面我们首先从张量算子 $\Delta^{(2n)}$ 开始介绍。详情可见文献 [2] 的第3章，这篇文章在scihub上可以直接下载。

格子张量

张量算子 $\Delta^{(2n)}$ 实际上计算的是将 $2n$ 个指标全部两两配对的所有方式之和。它的具体形式为：

$$\begin{aligned} \Delta^{(2)}_{ij}&=\delta_{ij} \\ \Delta^{(4)}_{ijkl}&=\delta_{ij}\delta_{kl} + \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} \\ \Delta^{(6)}_{ijklpq}&=\delta_{ij}\Delta^{(4)}_{ijkl} + \delta_{ik}\Delta^{(4)}_{jlpq} + \delta_{il}\Delta^{(4)}_{jkpq} + \delta_{ip}\Delta^{(4)}_{jklq} + \delta_{iq}\Delta^{(4)}_{jklp} \end{aligned}$$

并存在如下递归关系：

$$\Delta^{(2n)}_{i_1 i_2 \dots i_{2n}} = \sum_{j=2}^{2n} \delta_{i_1, i_{j}} \cdot \Delta^{(2n-2)}_{i_2 \dots i_{j-1} i_{j+1} \dots i_{2n}}$$

其中 $\delta_{ij}$ 为克罗内克积。

1. $2n$ 个指标分成 $n$ 对再求和，共有 (2n−1)!! 项。
   计算思路为 ：第一个元素有 (2n−1) 个选择配对；剩下 (2n−2) 个元素之中的第一个有 (2n−3) 个选择；后续依此类推。
2. 有的书会从排列组合视角出发，写成 $\frac{(2n)!}{2^n n!} = \frac{\text{总积}}{\text{偶数部分的积}}$，结果是等价的。
   计算思路为 ：$2n$ 个元素的全排列，除以每对内部的顺序($2n$)，再除以 $n$ 对之间的顺序($n!$)。

为了简洁表示 $\Delta^{(2n)}$ 的分量，文献 [2] 引入了 upper simplicial components，其索引是非递增的多重指标（如$(i_1,i_2,\dots,i_k)$，满足$i_1 \geq i_2 \geq \dots \geq i_k$）。 假设我们讨论一下二维空间的 $\Delta^{(4)}_{ijkl}$ ，其中的 $i,j,k,l$ 取 1 或 2。 那么下标的所有排列便是：

• 1111（全1，非递增）→ 值3
• 2111（1个2、3个1，非递增）→ 值0
• 2211（2个2、2个1，非递增）→ 值1
• 2221（3个2、1个1，非递增）→ 值0
• 2222（全2，非递增）→ 值3

即 $\Delta^{(4)}=[3,0,1,0,3]$.

在后续的描述中，为了避免写一大堆下标，我们采用如下简记

$$E^{(n)} = E^{(n)}_{i_1 i_2 \dots i_{n}}, \quad \Delta^{(2n)} = \Delta^{(2n)}_{i_1 i_2 \dots i_{2n}}, \quad \delta^{(2n)} = \delta^{(2n)}_{i_1 i_2 \dots i_{2n}},$$

这里的 $\Delta^{(2n)}$ 就是 $\{i_1, i_2, \dots i_{2n}\}$ 一系列下标组成的 $\Delta^{(2n)}_{i_1 i_2 \dots i_{2n}}$。而 $\delta^{(2n)}$ 则是2n个下标的克罗内克积，当所有下标相等时才等于1，否则为0。

速度矩与权重系数

从 $E^{(2n)}$ 的定义中，若假设所有 $\boldsymbol{e}_{\alpha}$ 一样长，可得：

$$\frac{1}{M} \sum_{\alpha} (\boldsymbol{e}_{\alpha} \cdot \boldsymbol{v})^{2n} = \underbrace{\frac{(2n-1)!!}{d (d+2) ... (d+2n-2)}}_{Q_{2n}} \|\boldsymbol{v}\|^{2n}$$

对于二维情况 ($d=2$)，则 $Q_2 = 1/2$, $Q_4 = 3/8$, $Q_6 = 5/16$, $Q_8 = 35/128$。对于三维情况 ($d=3$)，则 $Q_{2n} = 1 / (2n+1)$.

同理，拓展到奇数阶，则有：

$$\frac{1}{M} \sum_{\alpha} (\boldsymbol{e}_{\alpha} \cdot \boldsymbol{v})^{2n} (\boldsymbol{e}_{\alpha} \cdot \boldsymbol{v}) = Q_{2n} \|\boldsymbol{v}\|^{2n} \boldsymbol{v}$$

因此，对于一个所有 $\boldsymbol{e}_{\alpha}$ 一样长的离散速度集来说，它的离散速度矩的权重是确定的一个数 —— $Q_{2n}$。

假如 ${\boldsymbol{e}_{\alpha}}$ 中的所有速度并不等长，我们则可以认为：

$$E_{i_1 i_2 ... i_n}^{(n)} = \sum_{\alpha} \omega(\|\boldsymbol{e}_{\alpha}\|) \cdot (\boldsymbol{e}_{\alpha})_{i_1} (\boldsymbol{e}_{\alpha})_{i_2} ... (\boldsymbol{e}_{\alpha})_{i_n}$$

其中 $\omega(\|\boldsymbol{e}_{\alpha}\|)$ 指代由特征速度的模 $\|\boldsymbol{e}_{\alpha}\|$ 决定的权重系数。

DnQb 模型的确定

Qian等的 DnQb 模型采用如下的平衡态分布：

$$f^{eq}_{\alpha} (\rho, \boldsymbol{u}; \boldsymbol{e}_{\alpha}, \omega_{\alpha}) = \omega_{\alpha} \rho \cdot \left[ 1 + \frac{\boldsymbol{e}_{\alpha} \cdot \boldsymbol{u}}{c_s^2} + \frac{(\boldsymbol{e}_{\alpha} \cdot \boldsymbol{u})^2}{2 c_s^4} - \frac{\|\boldsymbol{u}\|^2}{2 c_s^2} \right].$$

其中 $\rho$ 为流场密度；$\boldsymbol{u}$ 为流场速度，$c_s^2 = RT$。若是为了模拟Navier-Stokes方程，平衡态分布函数需要满足下面的约束：

$$\begin{aligned} \sum_{\alpha} f^{eq}_{\alpha} &= \rho ,\\ \sum_{\alpha} f^{eq}_{\alpha} \boldsymbol{e}_{\alpha} &= \rho \boldsymbol{u} ,\\ \sum_{\alpha} f^{eq}_{\alpha} e_{\alpha i} e_{\alpha j} &= \rho u_i u_j + p \delta_{ij}. \end{aligned} \tag{2-2}$$

在待定系数法的框架下，就是把已经设计好的离散速度集 $\{\boldsymbol{e}_{\alpha}\}$ 代入上面的约束中，然后求解对应的权重 $\omega_{\alpha}$。

在这种视角下，我们要怎么确定就是要用 D2Q9、D3Q15、D3Q9 这些形式呢？

首先，正四边形最多能保证三阶格子张量是各向同性的，而到了四阶就变成了 $E_{ijkl}^{(4)}|_{M=4} = 2 \delta_{ijkl}$。为了解决这个问题，我们可以在在原有最近邻链接基础上，增加更远邻的链接，扩大速度方向集合 $\{\boldsymbol{e}_{\alpha}\}$。 而正方晶格加对角线的方式，就构成了典型的 D2Q9 模型：此时速度方向等效于正八边形顶点，当 $M=8>4$ 时，$E^{(4)}$ 满足各向同性条件（$M$ 不整除4）。

同样的道理，在三维空间中若是使用立方晶格，此时速度方向为立方体顶点（$M=8$，最近邻链接），对称性较低。$E^{(4)}$ 也是各向异性的，无法导出标准流体方程。此时也需要把一些对角线方向加进来，让 $E^{(4)}$ 满足各向同性条件。

Example: 以 D2Q9 为例

首先，我们把 D2Q9 分成三组

组别	速度配置	权重
1	$(0,0)$	$\omega_0$
2	cyc:$(\pm c,0)$	$\omega_1=\omega_2=\omega_3=\omega_4$
3	$(\pm c,\pm c)$	$\omega_5=\omega_6=\omega_7=\omega_8$

对于组别 2，根据格子张量的表格，我们有

$$\begin{aligned} \sum_{\alpha\in[1-4]} e_{\alpha i} &= 0 ,& \sum_{\alpha\in[1-4]} e_{\alpha i} e_{\alpha j} &= 2\delta_{ij} \cdot c^2\\ \sum_{\alpha\in[1-4]} e_{\alpha i} e_{\alpha j} e_{\alpha k} &= 0 ,& \sum_{\alpha\in[1-4]} e_{\alpha i} e_{\alpha j} e_{\alpha k} e_{\alpha l} &= 2\delta_{ijkl} \cdot c^4\\ \end{aligned}$$

对于组别 3，同理可得

$$\begin{aligned} \sum_{\alpha\in[5-8]} e_{\alpha i} &= 0 ,& \sum_{\alpha\in[5-8]} e_{\alpha i} e_{\alpha j} &= 2\delta_{ij} \cdot (\sqrt{2} c)^2\\ \sum_{\alpha\in[5-8]} e_{\alpha i} e_{\alpha j} e_{\alpha k} &= 0 ,& \sum_{\alpha\in[5-8]} e_{\alpha i} e_{\alpha j} e_{\alpha k} e_{\alpha l} &= (4 \Delta^{(4)}_{ijkl} - 8 \delta^{(4)}_{ijkl}) \cdot (\sqrt{2} c)^4\\ \end{aligned}$$

代入式(2-2)，依次得：

$$\rho (\omega_0 + 4 \omega_1 + 4 \omega_5) + \rho \|\boldsymbol{u}\|^2 \left[ \omega_1 \left(\frac{c^2}{c_s^4} - \frac{2}{c_s^2}\right) \right.\\ \left. + \omega_5 \left(\frac{2 c^2}{c_s^4} - \frac{2}{c_s^2}\right) - \frac{\omega_0}{2 c_s^2} \right] = \rho \tag{2-3}$$

$$\rho \boldsymbol{u} \left( \omega_1 \frac{2 c^2}{c_s^2} + \omega_5 \frac{4 c^2}{c_s^2} \right) = \rho \boldsymbol{u} \tag{2-4}$$

$$\begin{aligned} \rho \omega_1 \sum_{\alpha=1}^{4} \left[ e_{\alpha i} e_{\alpha j} \left(1 - \frac{u^2}{2 c_s^2}\right) + \frac{e_{\alpha i} e_{\alpha j} e_{\alpha k} e_{\alpha l} u_k u_l}{2 c_s^4} \right] &+ \\ \rho \omega_5 \sum_{\alpha=5}^{8} \left[ e_{\alpha i} e_{\alpha j} \left(1 - \frac{u^2}{2 c_s^2}\right) + \frac{e_{\alpha i} e_{\alpha j} e_{\alpha k} e_{\alpha l} u_k u_l}{2 c_s^4} \right] &= \rho u_i u_j + p \delta_{ij} \end{aligned} \tag{2-5}$$

由式(2-3)，得：

$$\omega_0 + 4 \omega_1 + 4 \omega_5 = 1 \\ \omega_1 \left(\frac{c^2}{c_s^4} - \frac{2}{c_s^2}\right) + \omega_2 \left(\frac{2 c^2}{c_s^4} - \frac{2}{c_s^2}\right) - \frac{\omega_0}{2 c_s^2} = 0$$

由式(2-4)，得：

$$(\omega_1 + 2 \omega_5) \frac{2 c^2}{c_s^2} =1$$

式(2-5)需要用到如下化简：

$$\begin{aligned} \Delta^{(4)}_{ijkl} u_{k} u_{l} &= \delta_{ij} (\delta_{kl} u_{k} u_{l}) + (\delta_{ik}\delta_{jl}u_{k} u_{l}) + (\delta_{il}\delta_{jk} u_{k} u_{l}) \\ &=\delta_{ij} \|\boldsymbol{u}\|^2 + 2 u_i u_j \end{aligned}$$

其中：

• 第一项仅当 $k=l$ 时不为0。因此 $\delta_{kl} u_k u_l = \sum_{k=l} u_k u_l = \sum_{k} u_k u_k = \|\boldsymbol{u}\|^2$。
• 第二项 $\delta_{ik}$ 要求 $k=i$ 时才非零，$\delta_{jl}$ 要求 $l=j$ 时才非零。因此对 $k,l$ 求和时，只有 $k=i$ 且 $l=j$ 的项保留，即 $u_i u_j$。
• 第三项 $\delta_{il}$ 要求 $l=i$，$\delta_{jk}$ 要求 $k=j$。因此只有 $l=i$ 且 $k=j$ 的项保留，即 $u_i u_j$。

由于式(2-5)的左侧可被简化为

$$\begin{aligned} (\omega_1 - 4 \omega_5) \frac{c^4}{c_s^4} \rho \delta_{ijkl} u_k u_l &+ \rho c^2 \delta_{ij} \left[ (1 - \frac{\|\boldsymbol{u}\|^2}{2 c_s^2}) (2\omega_1 + 4\omega_5) \right. \\ &\left. + 2 \omega_5 \frac{\|\boldsymbol{u}\|^2 c^2}{c_s^4} \right] + \rho u_i u_j w_5 \frac{4 c^2}{c_s^4} \tag{2-6} \end{aligned}$$

所以

$$\begin{aligned} w_5 \frac{4 c^2}{c_s^4} &= 1 \\ \omega_1 - 4 \omega_5 &= 0 \\ \left(1 - \frac{\|\boldsymbol{u}\|^2}{2 c_s^2}\right) (2\omega_1 + 4\omega_5) + 2 \omega_5 \frac{\|\boldsymbol{u}\|^2 c^2}{c_s^4} &= \frac{p}{\rho c^2} \end{aligned}$$

联立这些方程就可以解得： $\omega_0 = 4/9$, $\omega_1 = 1/9$, $\omega_5 = 1/36$, $c_s = c / \sqrt{3}$。

附录

（1）对于二维正M边形网格，$\boldsymbol{e}_{\alpha} = e \left( \cos(\frac{2\pi\alpha}{M}), \sin(\frac{2\pi\alpha}{M})\right)$。此时阶数 $n$ 与 $M$ 的对应关系为：

各向同性张量 $E^{(n)}$	正多边形边数 $M$
$E^{(2)}$	$M \gt 2$
$E^{(3)}$	$M \ge 2, M \neq 3$
$E^{(4)}$	$M \gt 2, M \neq 4$
$E^{(5)}$	$M \ge 2, M \neq [3,5]$
$E^{(6)}$	$M \gt 4, M \neq 6$
$E^{(7)}$	$M \ge 2, M \neq [3,5,7]$

（2）文献[2]给出了三维情况下 $E^{(n)}$ 的各向同性情况，下表用 Y 和 N 分别表示是和否。其中 cyc 表示循环排列, $\phi=(1+\sqrt{5})/2$。

几何形状	速度配置	M	$E^{(2)}$	$E^{(3)}$	$E^{(4)}$	$E^{(5)}$	$E^{(6)}$
四面体	$(1,1,1)$, cyc:$(1,-1,-1)$	4	Y	N	N	N	N
立方体	cyc:$(\pm 1,\pm 1,\pm 1)$	8	Y	Y	N	Y	N
八面体	cyc:$(\pm 1,0,0)$	6	Y	Y	N	Y	N
十二面体	$(\pm 1,\pm 1,\pm 1)$, cyc: $(0, \pm \phi-1, \pm \phi)$	20	Y	Y	Y	Y	N
二十面体	cyc: $(0, \pm \phi, \pm 1)$	12	Y	Y	Y	Y	N

（3）两种基本正方形格子模型的偶数阶张量

速度配置	M	$E^{(2)}$	$E^{(4)}$	$E^{(6)}$
cyc:$(\pm 1,0)$	4	$2 \delta^{(2)}$	$2 \delta^{(4)}$	$2 \delta^{(6)}$
$(\pm 1,\pm 1)$	8	$4 \delta^{(2)}$	$4 \Delta^{(4)} - 8 \delta^{(4)}$	$\frac{4}{3} \Delta^{(6)} - 16 \delta^{(6)}$

（4）几种最对称的三维Bravais晶格的偶数阶张量

形状	速度配置	M	$E^{(2)}$	$E^{(4)}$	$E^{(6)}$
Primitive cubic	cyc:$(\pm 1,0,0)$	6	$2 \delta^{(2)}$	$2 \delta^{(4)}$	$2 \delta^{(6)}$
Body-centered cubic	$(\pm 1,\pm 1,\pm 1)$	8	$8 \delta^{(2)}$	$8 (\Delta^{(4)} - 2\delta^{(4)})$	$8 (\Delta^{(6)} - 2 \Delta^{(4,2)} + 16\delta^{(6)})$
Face-centered cubic	cyc:$(\pm 1,\pm 1,0)$	12	$8 \delta^{(2)}$	$4 (\Delta^{(4)} - \delta^{(4)})$	$4 (\Delta^{(4,2)} + 13\delta^{(6)})$

在文献[2]的原话中：

$\Delta^{(n,m)}$ is the sum of all possible products of pairs of Kronecker delta symbols with $n$ and $m$ indices。

具体到 $\Delta^{(4,2)} = \Delta^{(4,2)}_{ijklmn}$ ，它应该是 $\mathrm{C}_{6}^{2} = 15$ 个项的和。打个比方：第1个项是 $\delta_{ij}\Delta^{(4)}_{klmn}$；第2个项是 $\delta_{ik}\Delta^{(4)}_{jlmn}$；后续依次类推，直到把15个组合都轮换出来，然后求和。

参考

[1] HE X, LUO L S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation[J]. Physical Review E, 1997, 56(6): 6811-6817. DOI:10.1103/PhysRevE.56.6811.
[2] WOLFRAM S. Cellular automaton fluids 1: Basic theory[J]. Journal of Statistical Physics, 1986, 45(3-4): 471-526. DOI:10.1007/BF01021083.
[3] 何雅玲, 王勇, 李庆, 童自翔. 格子Boltzmann方法的理论及应用[M]. 1 版. 北京: 高等教育出版社, 2023.

Lattice Boltzmann method

The (force-free) LBM equation is

$$f_{i}(\boldsymbol{x} + \boldsymbol{c}_{i}, t+\delta_t) - f_{i}(\boldsymbol{x}, t) = \Omega_{i}(\boldsymbol{x}, t).$$

In LBGK model, $\Omega_{i}(\boldsymbol{x}, t) = -\frac{1}{\tau} \left[ f_{i}(\boldsymbol{x}, t) - f_{i}^{(eq)} \right]$.

The $f_{i}^{(eq)}$ is the equilibrium state,

$$f_{i}^{(eq)} = w_{i} \rho \left[ 1 + \frac{\boldsymbol{c}_{i} \cdot \boldsymbol{u}}{c_s^2} + \frac{(\boldsymbol{c}_{i} \cdot \boldsymbol{u})^2}{2 c_s^4} - \frac{|\boldsymbol{u}|^2}{2 c_s^2} \right].$$

$w_i$ is the weight of the $\boldsymbol{c}_{i}$. $c_s$ is the speed of sound. $\rho, \boldsymbol{u}$ are the fluid density and velocity, respectively.

For simplicity, we define $f^{*}_{i} = f_{i} - \Omega_{i}$ as the post-collision population.

Curved boundary in LBM

NOTE: ① In the left figure, $\boldsymbol{x}_{w} = \boldsymbol{x}_{F} + q \boldsymbol{c}_{\bar{i}}$, and $\boldsymbol{c}_{\bar{i}} = -\boldsymbol{c}_{i}$. ② Let the yellow node is $\vec{x}_b$, then $q = |\vec{x}_w - \vec{x}_F| / |\vec{x}_b - \vec{x}_F|$

Interpolated bounce-back method

It treats the unknown $f_{i}(\boldsymbol{x}_{F}, t+\delta_t)$ as a polynomial of the known populations.

$$f_{i}(\boldsymbol{x}_{F}, t+\delta_t) = a_1 f_{\bar{i}}^{*}(\boldsymbol{x}_{FF}, t) + a_2 f_{\bar{i}}^{*}(\boldsymbol{x}_{F}, t) + a_3 f_{i}^{*}(\boldsymbol{x}_{F}, t) + K$$

where $K$ is a correction term.

|👉 How to find the coefficients $a_i$?
(1) Use Chapman-Enskog expansion, Taylor expansion, or Grad’s method to find the coefficients.
(2) BFL: a mesoscopic, geometrical approach proposed by Bouzidi et al.

BFL scheme

It determinates the boundary scheme based on the value of $q$. The length of the orange line is 1.

When $q \ge \frac{1}{2}$ (Figure (a)), we have

$$f_{i}(\boldsymbol{x}_{+}, t+\delta_t) = f_{\bar{i}}^{*} (\boldsymbol{x}_{F}, t) ,\quad f_{i}(\boldsymbol{x}_{FF}, t+\delta_t) = f_{i}^{*} (\boldsymbol{x}_{F}, t).$$

Then we can do linear interpolation to calculate $f_{i}(\boldsymbol{x}_{F}, t+\delta_t)$:

$$\begin{aligned} f_{i}(\boldsymbol{x}_{F}, t+\delta_t) =& f_{i}(\boldsymbol{x}_{+}, t+\delta_t) + \\ &\frac{2q - 1}{2q} [f_{i}(\boldsymbol{x}_{FF}, t+\delta_t)- f_{i}(\boldsymbol{x}_{+}, t+\delta_t)] \\ =& \frac{1}{2q} f_{\bar{i}}^{*} (\boldsymbol{x}_{F}, t) + (1-\frac{1}{2q}) f_{i}^{*} (\boldsymbol{x}_{F}, t) \end{aligned}$$

When $q < \frac{1}{2}$ (Figure (b)), we can do linear interpolation, i.e.,

$$\begin{aligned} f_{i}(\boldsymbol{x}_{F}, t+\delta_t) =& f_{\bar{i}}^{*}(\boldsymbol{x}_{+}, t)\\ =& f_{\bar{i}}^{*}(\boldsymbol{x}_{F}, t) \\ & + (1-2q) [f_{\bar{i}}^{*}(\boldsymbol{x}_{FF}, t) - f_{\bar{i}}^{*}(\boldsymbol{x}_{F}, t)]\\ =& 2q f_{\bar{i}}^{*}(\boldsymbol{x}_{F}, t) + (1-2q) f_{\bar{i}}^{*}(\boldsymbol{x}_{FF}, t) \end{aligned}$$

One-node local boundary

We can notice that: the general scheme usually employs the data of $\boldsymbol{x}_{FF}$, which is the second layer of fluid.

To design local linkwise boundary condition, one must discard the nonlocal contribution $f^{*}(\boldsymbol{x}_{FF}, t)$ that appears in the interpolation scheme.

Because the existence of $f^{*}(\boldsymbol{x}_{FF}, t)$ would make the scheme hard to describe the narrow region or corner.

1st order time approximation

The boundary node $\vec{x}_w$ is at the middle of $\vec{x}_1$ and $\vec{x}_2$, which is easier to use the half-way bounce-back (J. Fluid Mech., 271 (1994), pp. 285–309).

The first one is based on the following first-order in time approximation:

$$\underbrace{f_{\bar{i}}^{*}(\boldsymbol{x}_{FF}, t) = f_{\bar{i}}(\boldsymbol{x}_{F}, t+\delta_t) }_{\text{LBM streaming step}} \approx \underbrace{f_{\bar{i}}(\boldsymbol{x}_{F}, t)}_{\text{Approximation}}$$

Zhao et al.[3] point out that: their scheme can obtain second-order accuracy under the diffusive scaling ($\delta_t = O(\delta_x^2)$) [3].

Here:

$\vec{x}_w = \vec{x}_{F} + q \vec{c}_{\bar{i}}$, $\vec{x}_1 = \vec{x}_{F} + l \vec{c}_{\bar{i}}$, $\vec{x}_2 = 2 \vec{x}_{w} - \vec{x}_1$.

Stability condition: $l \in [\max\{0, 2q-1\}, 2q]$

With $\vec{x}_1$ and $\vec{x}_{FF}$, the interpolation of $\vec{x}_{F}$ is:

$$\begin{aligned} f_i (\vec{x}_{F}, t + \delta_t) =& \frac{l}{1+l} f_i (\vec{x}_{FF}, t + \delta_t) + \frac{1}{1+l} f_i (\vec{x}_{1}, t + \delta_t) \\ =& \underbrace{\frac{l}{1+l} f_i^{*} (\vec{x}_{F}, t)}_{\text{LBM Streaming}} + \frac{1}{1+l} f_i (\vec{x}_{1}, t + \delta_t) \end{aligned}$$

For the unknown distribution function $f_i (\vec{x}_{1}, t + \delta_t)$, Zhao et al.[3] use the half-way bounce-back purposed by Ladd (J. Fluid Mech., 271 (1994), pp. 285–309).

$$f_i (\vec{x}_{1}, t + \delta_t) = f_{\bar{i}} (\vec{x}_{2}, t + \delta_t) + 2 w_i \rho_0 \frac{\boldsymbol{c}_i \cdot \boldsymbol{u}(\vec{x}_b, t)}{c_s^2}$$

Here the wall velocity $\boldsymbol{u}(\vec{x}_b, t)$ is a known quantity.

Next, we interpolate $f_{\bar{i}} (\vec{x}_{2}, t + \delta_t)$ with the distribution functions at $\vec{x}_F$ and $\vec{x}_b$:

$$f_{\bar{i}} (\vec{x}_{2}, t + \delta_t) = (1+l-2q) f_{\bar{i}} (\vec{x}_{F}, t + \delta_t) + (2q-l) f_{\bar{i}} (\vec{x}_{b}, t + \delta_t)$$

Consider the streaming step from $\vec{x}_F$ and $\vec{x}_b$, we have: $f_{\bar{i}}^{*} (\vec{x}_{F}, t) = f_{\bar{i}} (\vec{x}_{b}, t + \delta_t)$.

Finally, with the approximation

$$\underbrace{f_{\bar{i}}^{*}(\boldsymbol{x}_{FF}, t) = f_{\bar{i}}(\boldsymbol{x}_{F}, t+\delta_t) }_{\text{LBM streaming step}} \approx \underbrace{f_{\bar{i}}(\boldsymbol{x}_{F}, t)}_{\text{Approximation}},$$

we can obtain the local boundary scheme:

$$f_i (\vec{x}_{F}, t + \delta_t) = \underbrace{ \frac{1+l-2q}{1+l} f_{\bar{i}} (\vec{x}_F, t) }_{\text{before collision}} + \underbrace{ \frac{l}{1+l} f_{i}^{*}(\boldsymbol{x}_{F}, t) + \frac{2q-l}{1+l} f_{\bar{i}}^{*}(\boldsymbol{x}_{F}, t) }_{\text{post collision terms}} + \underbrace{\frac{2 w_i \rho_0}{(1+l)c_s^2} \boldsymbol{c}_i \cdot \boldsymbol{u}(\vec{x}_b, t)}_{\text{Ladd's bounce-back}}$$

[NOTE]
(1) This scheme doesn’t rely on the collision model.
(2) If $\boldsymbol{x}_{FF}$ is always available, we can remove the first-order time approximation, and convert the scheme into a two-node scheme.
(3) When $l = q$, the scheme is equivalent to that purposed by Geier et al. (Comput. Math. Appl., 70 (2015), pp. 507–547)

$$f_i (\vec{x}_{F}, t + \delta_t) = \frac{1-q}{1+q} f_{\bar{i}} (\vec{x}_F, t) + \frac{q}{1+q} [f_{i}^{*}(\boldsymbol{x}_{F}, t) + f_{\bar{i}}^{*}(\boldsymbol{x}_{F}, t)] + \frac{2 w_i \rho_0}{(1+q)c_s^2} \boldsymbol{c}_i \cdot \boldsymbol{u}(\vec{x}_b, t)$$

If we still use $f_{\bar{i}}^{*} (\vec{x}_{FF}, t)$ rather than replacing it with $f_{\bar{i}} (\vec{x}_{F}, t)$, the scheme is equivalent to that purposed by Yu et al. (AIAA Paper, 2003-0953, 2003.).

Non-equilibrium state reconstruction

Tao et al.[4] have proposed a scheme to reconstruct the equilibrium and non-equilibrium state of the fluid node near the wall.

For the boundary node $\vec{x}_F$, its unknown distribution function can be calculated by the following interpolation:

$$f_{i}(\vec{x}_F, t + \delta_t) = \frac{1}{1+q} f_{i}(\vec{x}_w, t + \delta_t) + \frac{q}{1+q} f_{i}(\vec{x}_{FF}, t + \delta_t).$$

Then we begin to discuss the reconstruction process.

First, consider the streaming step from $\vec{x}_F$ to $\vec{x}_{FF}$, we have:

$$f_{i}^{*}(\vec{x}_{F}, t) = f_{i}(\vec{x}_{FF}, t + \delta_t)$$

Second, the term $f_{i}(\vec{x}_w, t + \delta_t)$ can be divided into two parts of equilibrium ($f_{i}^{(eq)}$) and non-equilibrium states ($f_{i}^{(neq)}$):

$$\begin{aligned} f_{i}(\vec{x}_w, t + \delta_t) &= f_{i}^{(eq)}(\vec{x}_w, t + \delta_t) + f_{i}^{(neq)}(\vec{x}_w, t + \delta_t) \\ &= f_{i}^{(eq)}(\vec{x}_w, t + \delta_t) + f_{i}^{(neq)}(\vec{x}_F, t + \delta_t) \\ &= f_{i}^{(eq)}(\vec{x}_w, t + \delta_t) + f_{\bar{i}}^{(neq)}(\vec{x}_F, t) \end{aligned}$$

Consider the substitution from $f_{i}^{(neq)}(\vec{x}_F, t + \delta_t)$ to $f_{\bar{i}}^{(neq)}(\vec{x}_F, t)$ as a non-equilibrium bounce-back, as mentioned in Tao et al.[4].

The $f_{i}^{(eq)}(\vec{x}_w, t + \delta_t)$ can be determined by the known velocity and the approximate fluid density

$$\begin{aligned} f_{i}^{(eq)}(\vec{x}_w, t + \delta_t) &= f_{i}^{(eq)} \left( \vec{u}(\vec{x}_w, t + \delta_t), \rho(\vec{x}_w, t + \delta_t) \right) \\ &\approx f_{i}^{(eq)} \left( \vec{u}(\vec{x}_w, t + \delta_t), \rho(\vec{x}_F, t + \delta_t) \right) \\ &\approx f_{i}^{(eq)} \left( \vec{u}(\vec{x}_w, t + \delta_t), \rho(\vec{x}_F, t) \right) \\ \end{aligned}$$

Here we cite the original sentences from the paper [4]:

It has been demonstrated that for low speed flow, using $\rho_A (t)$ and $\rho_A (t+\delta_t)$ to approximate $\rho_A (t+\delta_t)$ and $\rho_W (t+\delta_t)$ have second- and third-order accuracies, respectively [23,39].

== Reference ==
[23] Z. Guo, C. Zheng, B. Shi, An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids 14 (6) (2002) 2007–2010.
[39] Z. Guo, C. Shu, Lattice Boltzmann Method and Its Applications in Engineering, World Scientific, 2013.

So the boundary scheme purposed by Tao et al. [4] is:

$$f_{i}(\vec{x}_F, t + \delta_t) = \frac{q}{1+q} f_{i}^{*}(\vec{x}_{F}, t) + \frac{1}{1+q} \left\{ f_{i}^{(eq)} \left( \vec{u}(\vec{x}_w, t + \delta_t), \rho(\vec{x}_F, t) \right) + f_{\bar{i}}^{(neq)}(\vec{x}_F, t) \right\}$$

[NOTE]
This scheme sounds easy and familar, right? Let’s make a summary.
(1) It comes from the interpolation scheme.
(2) Employ streaming step to eliminate the unknown $f_{i}(\vec{x}_{FF}, t + \delta_t)$.
(3) The distribution function at the wall node $\vec{x}_w$ is divieded into two parts: $f_{i}^{(eq)}$ and $f_{i}^{(neq)}$.
(4) $f_{i}^{(eq)}$ is approximated by known macroscopic variables.
(5) $f_{i}^{(neq)}$ is approximated by the non-equilibrium bounce-back.

Enhanced single-node boundary condition

Here, we introduce the enhanced single-node boundary condition purposed by Marson et al.[1].

$$\begin{aligned} f_{i}(\vec{x}_F, t + \delta_t) =& \cancel{a_1 f_{\bar{i}}^{*}(\vec{x}_{FF})} + a_2 \underbrace{(\vec{x}_{F} + 2 f^{(eq)-}(\vec{x}_{w}))}_{\text{1}} + a_3 f_{i}^{*}(\vec{x}_{F})\\ & + a_4 \underbrace{ \tilde{f}_{i}(\vec{x}_w, t + \delta_t) }_{\text{2}} + a_5 \underbrace{\tilde{f}_{i}^{*}(\vec{x}_w)}_{\text{3}} - \underbrace{K^{-} \frac{f_{i}^{(neq)-}(\vec{x}_F)}{\tau^{-}}}_{\text{4}} \end{aligned}$$

Term ①: The haly-way bounce-back from Ladd (J. Fluid Mech., 271 (1994), pp. 285–309).
Note that: $f^{(eq)-}(\vec{x}_{w}) = w_i \rho \vec{c}_i \cdot \vec{u}_w / c_s^2$.
Term ②: The term from [4].
Term ③: The new term designed by Marson et al.[1].
Term ④: The new (optional) term mentioned in the Sec. ⅣB of Marson et al.[1].

Marson et al.[1] use TRT collision model. So here we briefly introduce the two-relaxation-time (TRT) collision model.

$$\begin{cases} f_{i}^{+} (\vec{x} + \vec{c}_i \delta_t, t + \delta_t) - f_{i}^{+} (\vec{x}, t) = -\frac{1}{\tau^{+}} (f_{i}^{+} (\vec{x}, t) - f_{i}^{(eq)+} (\vec{x}, t)) \\ f_{i}^{-} (\vec{x} + \vec{c}_i \delta_t, t + \delta_t) - f_{i}^{-} (\vec{x}, t) = -\frac{1}{\tau^{-}} (f_{i}^{-} (\vec{x}, t) - f_{i}^{(eq)-} (\vec{x}, t)) \\ \end{cases}$$

where

$$\begin{aligned} f_{i}^{+} = \frac{1}{2} (f_i + f_{-i}) ,\quad f_{i}^{-} = \frac{1}{2} (f_i - f_{-i}) ,\\ f_{i}^{(eq)+} = \frac{1}{2} (f_i^{(eq)} + f_{-i}^{(eq)}) = w_i \rho (1 + \frac{(\vec{c}_i \cdot \vec{u})^2}{2 c_s^4} - \frac{|\vec{u}|^2}{2 c_s^2}) ,\\ f_{i}^{(eq)-} = \frac{1}{2} (f_i^{(eq)} - f_{-i}^{(eq)}) = w_i \rho \frac{\vec{c}_i \cdot \vec{u}}{c_s^2} ,\\ \end{aligned}$$

and the TRT magic number is: $\Lambda = (\tau^{+} - \frac{1}{2}) (\tau^{-} - \frac{1}{2})$.

Marson et al.[1] design the wall populations $\tilde{f}_{i}^{*}(\vec{x}_w)$ and $\tilde{f}_{i}(\vec{x}_w, t+\delta_t)$.【$\Omega$ is the collision operator.】

$$\begin{aligned} \tilde{f}_{i}^{*}(\vec{x}_w, t) \overset{\text{def}}{=}& \tilde{f}_{i}^{(eq)}(\vec{x}_w, t) + (1 - \Omega) \tilde{f}_{i}^{(neq)}(\vec{x}_w, t) \\ \tilde{f}_{i}(\vec{x}_w, t+\delta_t) \overset{\text{def}}{=}& \tilde{f}_{i}^{(eq)}(\vec{x}_w, t+\delta_t) + \tilde{f}_{i}^{(neq)}(\vec{x}_w, t+\delta_t) \end{aligned}$$

Consider $\tilde{f}_{i}(\vec{x}_w, t+\delta_t)$, its equilibrium part is approximated by

$$\tilde{f}_{i}^{(eq)} (\vec{x}_w, t+\delta_t) \approx f_{i}^{(eq)+} (\rho(\vec{x}_F, t), \vec{u}(\vec{x}_{\{w, F\}}, t+\delta_t)) + f_{i}^{(eq)-} (\rho(\vec{x}_F, t), \vec{u}(\vec{x}_{w}, t+\delta_t))$$

Here, $\vec{x}_{\{w, F\}}$ means use $\vec{x}_{w}$ or $\vec{x}_F$ as the position.

The non-equilibrium part is approximated by an nonequilibrium bounce-back. This is a first-order approximation of the nonequilibrium bounce-back method of Zou and He, and it leads to the following second-order accurate approximation:

$$\tilde{f}_{i}^{(neq)}(\vec{x}_w, t+\delta_t) = \tilde{f}_{i}^{(neq)}(\vec{x}_F, t)$$

For the $\tilde{f}_{i}^{*}(\vec{x}_w, t)$ , we have

$$\tilde{f}_{i}^{(eq)} (\vec{x}_w, t) \approx f_{i}^{(eq)+} (\rho(\vec{x}_F, t), \vec{u}(\vec{x}_{\{w, F\}}, t)) + f_{i}^{(eq)-} (\rho(\vec{x}_F, t), \vec{u}(\vec{x}_{w}, t))$$

The non-equilibrium term is also split into symmetric and anti-symmetric parts: $\tilde{f}_{i}^{(neq)}(\vec{x}_w, t) = f_{i}^{(neq)+}(\vec{x}_w, t) + f_{i}^{(neq)-}(\vec{x}_w, t)$. Marson et al.[1] gives three ways to approximate the non-equilibrium term.

(1) Non-symmetric enhanced local interpolation (Denoted as N):

$$f_{i}^{(neq)+}(\vec{x}_w, t) \approx f_{i}^{(neq)+}(\vec{x}_F, t) ,\quad f_{i}^{(neq)-}(\vec{x}_w, t) \approx -f_{i}^{(neq)-}(\vec{x}_F, t)$$

(2) Symmetric enhanced local interpolation (Denoted as S):

$$f_{i}^{(neq)+}(\vec{x}_w, t) \approx f_{i}^{(neq)+}(\vec{x}_F, t) ,\quad f_{i}^{(neq)-}(\vec{x}_w, t) \approx f_{i}^{(neq)-}(\vec{x}_F, t)$$

(3) Central enhanced local interpolation (Denoted as C):

$$f_{i}^{(neq)+}(\vec{x}_w, t) \approx f_{i}^{(neq)+}(\vec{x}_F, t) ,\quad f_{i}^{(neq)-}(\vec{x}_w, t) = 0$$

The following table shows the interpolation coefficients provided by Marson et al. [1]. It should be noted that $K^{-}$ is the term used to eliminate viscosity errors in steady-state conditions. In practice, $\tau^{+} \rightarrow \frac{1}{2}$ at high Reynolds numbers, making the contribution of $K^{-} \frac{f_{i}^{(neq)-}(\vec{x}_F)}{\tau^{-}}$ less significant.

[NOTE]: The calculation of $K^{-}$ could be found in Marson et al. [1].

References

[1] Marson, F., Thorimbert, Y., Chopard, B., Ginzburg, I. & Latt, J. Enhanced single-node lattice Boltzmann boundary condition for fluid flows. Phys. Rev. E 103, 053308 (2021).
[2] 郭照立 & 郑楚光. 格子Boltzmann方法的原理及应用. (科学出版社, 2009).
[3] Zhao, W., Huang, J. & Yong, W.-A. Boundary conditions for kinetic theory based models I: Lattice boltzmann models. Multiscale Modeling & Simulation 17, 854–872 (2019).
[4] Tao, S., He, Q., Chen, B., Yang, X. & Huang, S. One-point second-order curved boundary condition for lattice Boltzmann simulation of suspended particles. Computers & Mathematics with Applications 76, 1593–1607 (2018).