In $\mathbb{R}^2$, the position and velocity vectors are given by $\vx=(\x,\y)^T$ and $\vu=(\ou,\ov)^T$, respectively.


\subsection{Discrete velocity vectors}
Discrete velocity vectors and the lattice speed of sound are defined by 
\input{\path/d2q5_C}
$$c_s = \frac1{{\sqrt{3}}},$$
respectively \cite{kruger}.

%\begin{figure}[htb!]
%	\centering
\begin{center}
	\includegraphics[width=0.5\textwidth]{\path/d2q5}
\end{center}
%	\caption{Illustration of the discrete d2q5 velocity model}
%	\label{xX}
%\end{figure}

\subsection{Raw and central moments}
The raw and central moments are defined by
\begin{equation*}
	m_\va := \sum_{i=1}^{5} f_i \vc_i^\va,
\end{equation*}
and
\begin{equation*}
	k_\va := \sum_{i=1}^{5} f_i (\vc_i - \vu)^\va,
\end{equation*}
respectively, where $\va=(\alpha_1,\alpha_2)\in\mathbb{Z}^2$ denotes a multi-index (as a row vector) and $\vc_i^\va := \prod_{j=1}^2 [\vc_i]_j^{\alpha_j}$.


\subsection{Transformation matrix $\matr{M}$}
Matrix $\matr{M}$, that defines macroscopic quantities (moments) $\vec{\mu}$ by 
$$
	\vec{\mu} = \matr{M} \vec{f},
$$
with $\vec{f} = \left( f_1, f_2, \dots,f_{5} \right)^T$,
is selected such that
\input{\path/d2q5_rawMom}
%$m_{\vec{\alpha}}$ are the raw moments defined by
%\begin{equation*}
%	m_\va := \sum_{i=1}^{5} f_i \vc_i^\va,
%\end{equation*}
i.e., $\matr{M}$ is given by
\input{\path/d2q5_matM}

\subsection{Equilibrium}

The corresponding equilibrium raw moments are defined using the continuous Maxwell--Boltzmann distribution function \cite{kruger} 
\begin{equation*}
f^{(eq)}(\vec{\xi}) = \frac{\rho}{2\pi c_s^2} \exp\left(-\frac{|| \vec{\xi} - \vec{v} ||^2}{2 c_s^2} \right)
\end{equation*}
as
\begin{equation*}
m^{(eq)}_{\vec{\alpha}} = \int\limits_{\mathbb{R}^2} \vec{\xi}^{\vec{\alpha}} f^{(eq)}(\vec{\xi}) \ud \vec{\xi},
\end{equation*}
where $\alpha_i \in \{0,1,2\}$, $i=1,2$. 
Hence, the equilibrium moments  $\vec{\mu}^{(eq)}$ satisfy
\input{\path/d2q5_eq}