In $\mathbb{R}^1$, the position and velocity vectors are given by $\vx=(\x)$ and $\vu=(\ou)$, respectively.


\subsection{Discrete velocity vectors}
Discrete velocity vectors and the lattice speed of sound are defined by 
\input{\path/d1q3_C}
$$c_s = \frac1{{\sqrt{3}}},$$
respectively \cite{kruger}.

%\begin{figure}[htb!]
%	\centering
\begin{center}
	\includegraphics[width=0.5\textwidth]{\path/d1q3}
\end{center}
%	\caption{Illustration of the discrete d1q3 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}^{3} f_i \vc_i^\va,
\end{equation*}
and
\begin{equation*}
	k_\va := \sum_{i=1}^{3} f_i (\vc_i - \vu)^\va,
\end{equation*}
respectively, where $\va=(\alpha_1)\in\mathbb{Z}^1$ denotes a multi-index and $\vc_i^\va :=  [\vc_i]_1^{\alpha_1}$.

\subsection{Transformation matrix $\matr{M}$}
Matrix $\matr{M}$, that defines macroscopic quantities (moments) $\vec{\mu}$ by 
$$
	\vec{\mu} = \matr{M} \vec{f},
$$
with $\vec{f} = \left( f_1, f_2, f_3 \right)^T$,
is selected such that
\input{\path/d1q3_rawMom}
%$m_{\vec{\alpha}}$ are the raw moments defined by
%\begin{equation*}
%	m_\va := \sum_{i=1}^{3} f_i \vc_i^\va,
%\end{equation*}
i.e., $\matr{M}$ is given by
\input{\path/d1q3_matM}

\subsection{Equilibrium}

The corresponding equilibrium raw moments are defined using the continuous Maxwell--Boltzmann distribution function \cite{kruger} 
\begin{equation*}
f^{(eq)}(\xi) = \frac{\rho}{\left(2 \pi c_s^2 \right)^{\frac{1}{2}}} \exp\left(-\frac{\left(\xi - \ou \right)^2}{2 c_s^2} \right)
\end{equation*}
as
\begin{equation*}
m^{(eq)}_{{(\alpha)}} = \int\limits_{\mathbb{R}} \xi^{\alpha} f^{(eq)}(\xi) \ud \xi ,
\end{equation*}
where $\alpha \in \{0,1,2\}$. 
Hence, the equilibrium moments  $\vec{\mu}^{(eq)}$ satisfy
\input{\path/d1q3_eq}


