Commit bd71c9a1 authored by Matthieu Schaller's avatar Matthieu Schaller
Browse files

Finished description of PE-SPH in the documentation

parent 57f7680e
......@@ -305,17 +305,72 @@ P_{\partial h_i}$ and $\rho_{\partial h_i}$
(eq. \ref{eq:sph:minimal:rho_dh}):
\begin{equation}
f_i \equiv \left(\frac{\bar P_{\partial h_i} h_i}{3\rho_i
\tilde{A}_i}\right)\left(1 + \frac{h_i}{3\rho_i}\rho_{\partial
h_i}\right)^{-1}.
f_i \equiv \left(\frac{h_i}{3\rho_i}\bar P_{\partial
h_i}\right)\left(1 + \frac{h_i}{3\rho_i}\rho_{\partial
h_i}\right)^{-1}.
\end{equation}
\subsubsection{Hydrodynamical accelerations (\nth{2} neighbour loop)}
The accelerations are given by the following term:
\begin{align}
\frac{d\vec{v}_i}{dt} = -\sum_j m_j &\left[\frac{\bar P_i}{\bar\rho_i^2} \left(\frac{\tilde A_j}{\tilde A_i} - \frac{f_i}{\tilde A_i}\right)\nabla_x W(\vec{x}_{ij}, h_i) \right. \nonumber \\
&+\frac{P_j}{\rho_j^2} \left(\frac{\tilde A_i}{\tilde A_j} - \frac{f_j}{\tilde A_j}\right)\nabla_x W(\vec{x}_{ij},h_j) \\
&+ \left. \bigg.\nu_{ij} \Wij \right], \label{eq:sph:pe:dv_dt}
\end{align}
where the viscosity term $\nu_{ij}$ has been computed as in
the \GadgetSPH case (Eq. \ref{eq:sph:gadget2:balsara}
and \ref{eq:sph:gadget2:nu_ij}). For completeness, the equation of
motion for the entropy is
\begin{equation}
\frac{dA_i}{dt} = \frac{1}{2} A_{\rm eos}\left(\rho_i, \sum_j
m_j \nu_{ij}\vec{v}_{ij}\cdot \Wij\right).
\end{equation}
\subsubsection{Time integration}
The time-step condition is identical to the \MinimalSPH case
(Eq. \ref{eq:sph:minimal:dt}). The same applies to the integration
forward in time (Eq. \ref{eq:sph:minimal:kick_v} to
\ref{eq:sph:minimal:kick_c}) with the exception of the change in
internal energy (Eq. \ref{eq:sph:minimal:kick_u}) which gets replaced
by an integration for the the entropy:
\begin{align}
\vec{v}_i &\rightarrow \vec{v}_i + \frac{d\vec{v}_i}{dt} \Delta t \label{eq:sph:pe:kick_v}\\
A_i &\rightarrow A_i + \frac{dA_i}{dt} \Delta t \label{eq:sph:pe:kick_A}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, A_i\right) \label{eq:sph:pe:kick_P}, \\
c_i &\rightarrow c_{\rm eos}\left(\rho_i,
A_i\right) \label{eq:sph:pe:kick_c}, \\
\tilde A_i &= A_i^{1/\gamma}
\end{align}
where, once again, we made use of the equation of state relating
thermodynamical quantities.
\subsubsection{Particle properties prediction}
The prediction step is also identical to the \MinimalSPH case with the
entropic function replacing the thermal energy.
\begin{align}
\vec{x}_i &\rightarrow \vec{x}_i + \vec{v}_i \Delta t \label{eq:sph:pe:drift_x} \\
h_i &\rightarrow h_i \exp\left(\frac{1}{h_i} \frac{dh_i}{dt}
\Delta t\right), \label{eq:sph:pe:drift_h}\\
\rho_i &\rightarrow \rho_i \exp\left(-\frac{3}{h_i} \frac{dh_i}{dt}
\Delta t\right), \label{eq:sph:pe:drift_rho} \\
\tilde A_i &\rightarrow \left(A_i + \frac{dA_i}{dt}
\Delta t\right)^{1/\gamma} \label{eq:sph:pe:drift_A_tilde}, \\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, A_i + \frac{dA_i}{dt} \Delta t\right), \label{eq:sph:pe:drift_P}\\
c_i &\rightarrow c_{\rm eos}\left(\rho_i, A_i + \frac{dA_i}{dt}
\Delta t\right) \label{eq:sph:pe:drift_c},
\end{align}
where, as above, the last two updated quantities are obtained using
the pre-defined equation of state. Note that the entropic function $A_i$
itself is \emph{not} updated.
\subsection{Pressure-Energy SPH}
\label{sec:sph:pu}
......
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