\subsection{Cosmological factors for properties entering the artificial viscosity}
\label{ssec:artificialvisc}

There are multiple properties that enter into the more complex artificial
viscosity schemes, such as those by \citet{Morris1997} (henceforth M\&M) and
\citet{Cullen2010} (henceforth C\&D).

\subsubsection{M\&M basic scheme}
\label{sssec:mandm}

This relies on the velocity divergence as a shock indicator, i.e. the property
$\nabla \cdot \mathbf{v}$. The interpretation of this is the velocity divergence of
the fluid overall, i.e. the physical velocity divergence. Starting with
\begin{equation}
\mathbf{v}_p = a \dot{\mathbf{r}}' + \dot{a} \mathbf{r}', \nonumber
\end{equation}
with the divergence,
\begin{equation}
\nabla \cdot \mathbf{v}_p =
    \nabla \cdot \left(a \dot{\mathbf{r}}'\right) +
    \nabla \cdot \left(\dot{a} \mathbf{r}'\right). \nonumber
\end{equation}
The quantity on the left is the one that we want to enter the source term for the
artificial viscosity. Transforming to the co-moving derivative on the right hand side
to enable it to be calculated in the code,
\begin{equation}
\nabla \cdot \mathbf{v}_p = 
    \nabla' \cdot \dot{\mathbf{r}}' + n_d H(a),
\label{eqn:divvwithcomovingcoordinates}
\end{equation}
with $n_d$ the number of spatial dimensions, and the final transformation
being the one to internal code velocity units,
\begin{equation}
\nabla \cdot \mathbf{v}_p = 
    \frac{1}{a^2} \nabla' \cdot \mathbf{v}' + n_d H(a).
\label{eqn:divvcodeunits}
\end{equation}
We note that there is no similar hubble flow term in the expression for
$\nabla \times \mathbf{v}_p$.

In some more complex schemes, such as the one presented by \cite{Cullen2010},
the time differential of the velocity divergence is used as a way to differentiate
the pre- and post-shock region.

Building on the above, we take the time differential of both sides,
\begin{equation}
    \frac{{\mathrm d}}{{\mathrm d} t} \nabla \cdot \mathbf{v}_p = 
    \frac{{\mathrm d}}{{\mathrm d} t} \left(
    	\frac{1}{a^2} \nabla' \cdot \mathbf{v}' + n_d H(a)
    \right).
    \nonumber
\end{equation}
Collecting the factors, we see
\begin{align}
    \frac{{\mathrm d}}{{\mathrm d} t} \nabla \cdot \mathbf{v}_p = 
    \frac{1}{a^2} &\left(
    	\frac{{\mathrm d}}{{\mathrm d} t} \nabla ' \cdot \mathbf{v}' -
    	2H(a) \nabla' \cdot \mathbf{v}'
    \right) \\
    + n_d &\left(
    	\frac{\ddot{a}}{a} - \frac{\dot{a}}{a^2}
    \right).
    \label{eqn:divvdtcodeunits}
\end{align}
This looks like quite a mess, but in most cases we calculate this implicitly
from the velocity divergence itself, and so we do not actually need to take
into account these factors; i.e. we actually calculate
\begin{equation}
    \frac{\mathrm d}{{\mathrm d} t} \nabla \cdot \mathbf{v}_p = 
    \frac{
    	\nabla \cdot \mathbf{v}_p (t + {\mathrm d}t) - \nabla \cdot \mathbf{v}_p (t)
    }{dt},
	\label{eqn:divvdtcodeunitsimplicit}
\end{equation}
meaning that the above is taken into account self-consistently.