Commit 4616fdaa authored by Matthieu Schaller's avatar Matthieu Schaller
Browse files

Finished description of density loop in PE-SPH

parent 85dd55f6
......@@ -411,8 +411,8 @@ __attribute__((always_inline)) INLINE static void hydro_end_force(
p->force.h_dt *= p->h * hydro_dimension_inv;
p->entropy_dt *=
0.5f * hydro_gamma_minus_one * pow_minus_gamma_minus_one(p->rho);
p->entropy_dt =
0.5f * gas_entropy_from_internal_energy(p->rho, p->entropy_dt);
}
/**
......
......@@ -77,6 +77,24 @@ archivePrefix = "arXiv",
}
@ARTICLE{Schaller2015,
author = {{Schaller}, M. and {Dalla Vecchia}, C. and {Schaye}, J. and
{Bower}, R.~G. and {Theuns}, T. and {Crain}, R.~A. and {Furlong}, M. and
{McCarthy}, I.~G.},
title = "{The EAGLE simulations of galaxy formation: the importance of the hydrodynamics scheme}",
journal = {\mnras},
archivePrefix = "arXiv",
eprint = {1509.05056},
keywords = {methods: numerical, galaxies: clusters: intracluster medium, galaxies: formation, cosmology: theory},
year = 2015,
month = dec,
volume = 454,
pages = {2277-2291},
doi = {10.1093/mnras/stv2169},
adsurl = {http://adsabs.harvard.edu/abs/2015MNRAS.454.2277S},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
......@@ -9,7 +9,11 @@
\newcommand{\dd}[2]{\frac{\partial #1}{\partial #2}}
\renewcommand{\vec}[1]{{\mathbf{#1}}}
\newcommand{\Wij}{\overline{\nabla_xW_{ij}}}
\newcommand{\tbd}{\textcolor{red}{TO BE DONE}}
\newcommand{\MinimalSPH}{\textsc{minimal-sph}\xspace}
\newcommand{\GadgetSPH}{\textsc{gadget-sph}\xspace}
\newcommand{\PESPH}{\textsc{pe-sph}\xspace}
%\setlength{\mathindent}{0pt}
%opening
......@@ -27,17 +31,19 @@ average over two particles of a quantity: $\bar a_{ij} \equiv
vector quantity $\Wij \equiv \frac{1}{2}\left(W(\vec{x}_{ij}, h_i) +
\nabla_x W(\vec{x}_{ij},h_j)\right)$.
\subsection{Minimal SPH}
%#########################################################################################################
\subsection{\MinimalSPH}
\label{sec:sph:minimal}
This is the simplest fully-conservative version of SPH using the
internal energy $u$ as a thermal variable that can be written
down. Its implementation in \swift should be understood as a text-book
example and template for more advanced implementation. A full
derivation of the equations can be found in the review of
\cite{Price2012}. Our implementation follows their equations (27), (43),
(44), (45), (101), (103) and (104) with $\beta=3$ and $\alpha_u=0$. We
summarize the equations here.
example and template for more advanced implementations. A full
derivation and motivation for the equations can be found in the review
of \cite{Price2012}. Our implementation follows their equations (27),
(43), (44), (45), (101), (103) and (104) with $\beta=3$ and
$\alpha_u=0$. We summarize the equations here.
\subsubsection{Density and other fluid properties (\nth{1} neighbour loop)}
......@@ -155,6 +161,7 @@ Inactive particles need to have their quantities predicted forward in
time in the ``drift'' operation. We update them as follows:
\begin{align}
\vec{x}_i &\rightarrow \vec{x}_i + \vec{v}_i \Delta t \label{eq:sph:minimal:drift_x} \\
h_i &\rightarrow h_i \exp\left(\frac{1}{h_i} \frac{dh_i}{dt}
\Delta t\right), \label{eq:sph:minimal:drift_h}\\
\rho_i &\rightarrow \rho_i \exp\left(-\frac{3}{h_i} \frac{dh_i}{dt}
......@@ -167,6 +174,8 @@ where, as above, the last two updated quantities are obtained using
the pre-defined equation of state. Note that the thermal energy $u_i$
itself is \emph{not} updated.
%#########################################################################################################
\subsection{Gadget-2 SPH}
\label{sec:sph:gadget2}
......@@ -182,10 +191,10 @@ presentation of \cite{Springel2005}. Specifically, we use their equations (5), (
For a set of particles $i$ with positions $\vec{x}_i$ with velocities
$\vec{v}_i$, masses $m_i$, entropic function per unit mass $A_i$ and
smoothing length $h_i$, we compute the density, derivative of the density with respect
to $h$ and the ``f-terms'' in a similar way to the minimal-SPH case
to $h$ and the ``h-terms'' in a similar way to the minimal-SPH case
(Eq. \ref{eq:sph:minimal:rho}, \ref{eq:sph:minimal:rho_dh} and
\ref{eq:sph:minimal:f_i}). From these the pressure and sound-speed can
be computed using the predefined equation of state:
be computed using the pre-defined equation of state:
\begin{align}
P_i &= P_{\rm eos}(\rho_i, A_i), \label{eq:sph:gadget2:P}\\
......@@ -195,9 +204,9 @@ We additionally compute the divergence and
curl of the velocity field using standard SPH expressions:
\begin{align}
\nabla\cdot\vec{v}_i \equiv\nabla\cdot \vec{v}(\vec{x}_i) &= \frac{1}{\rho_i} \sum_j m_j
\nabla\cdot\vec{v}_i &\equiv\nabla\cdot \vec{v}(\vec{x}_i) = \frac{1}{\rho_i} \sum_j m_j
\vec{v}_{ij}\cdot\nabla_x W(\vec{x}_{ij}, h_i) \label{eq:sph:gadget2:div_v},\\
\nabla\times\vec{v}_i \equiv \nabla\times \vec{v}(\vec{x}_i) &= \frac{1}{\rho_i} \sum_j m_j
\nabla\times\vec{v}_i &\equiv \nabla\times \vec{v}(\vec{x}_i) = \frac{1}{\rho_i} \sum_j m_j
\vec{v}_{ij}\times\nabla_x W(\vec{x}_{ij}, h_i) \label{eq:sph:gadget2:rot_v}.
\end{align}
These are used to construct the \cite{Balsara1995} switch $B_i$:
......@@ -223,16 +232,19 @@ whilst equations \ref{eq:sph:minimal:v_sig},
\ref{eq:sph:minimal:v_sig_update} remain unchanged. The only other
change is the equation of motion for the thermodynamic variable which
now has to be describing the evolution of the entropic function and
not the evolution of the thermal energy. It reads:
not the evolution of the thermal energy. Instead of
eq. \ref{eq:sph:minimal:du_dt}, we have
\begin{equation}
\frac{dA_i}{dt} = \frac{1}{2} \frac{\gamma-1}{\rho^{\gamma-1}} \sum_j
m_j \nu_{ij}\vec{v}_{ij}\cdot \Wij
\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}
where we made use of the pre-defined equation of state relating
density and internal energy to entropy.
\subsubsection{Time integration}
The time-step condition is identical to the Minimal-SPH case
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
......@@ -247,14 +259,15 @@ by an integration for the the entropy:
c_i &\rightarrow c_{\rm eos}\left(\rho_i, A_i\right) \label{eq:sph:gadget2:kick_c},
\end{align}
where, once again, we made use of the equation of state relating
thermodynaimcal quantities.
thermodynamical quantities.
\subsubsection{Particle properties prediction}
The prediction step is also identical to the Minimal-SPH case with the
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:gadget2:drift__x} \\
h_i &\rightarrow h_i \exp\left(\frac{1}{h_i} \frac{dh_i}{dt}
\Delta t\right), \label{eq:sph:gadget2:drift_h}\\
\rho_i &\rightarrow \rho_i \exp\left(-\frac{3}{h_i} \frac{dh_i}{dt}
......@@ -268,27 +281,76 @@ the pre-defined equation of state. Note that the entropic function $A_i$
itself is \emph{not} updated.
%#########################################################################################################
\subsection{Pressure-Entropy SPH}
\cite{Hopkins2013}
\label{sec:sph:pe}
\subsubsection{Hydrodynamical accelerations}
This flavour of SPH follows the implementation described in section
2.2.3 of \cite{Hopkins2013}. We start with their equations (17), (19),
(21) and (22) but modify them for efficiency and generality
reasons. We also use the same \cite{Balsara1995} viscosity switch as
in the \GadgetSPH scheme (Sec. \ref{sec:sph:gadget2}).
\subsubsection{Density and other fluid properties (\nth{1} neighbour loop)}
For a set of particles $i$ with positions $\vec{x}_i$ with velocities
$\vec{v}_i$, masses $m_i$, entropic function per unit mass $A_i$ and
smoothing length $h_i$, we compute the density, derivative of the
density with respect to $h$, divergence and curl of velocity field in
a similar fashion to the \GadgetSPH scheme. From the basic particle
properties we construct an additional temporary quantity
\begin{equation}
\tilde{A_i} \equiv A_i^{1/\gamma},
\label{eq:sph:pe:A_tilde}
\end{equation}
which enters many equations. This allows us to construct the
entropy-weighted density $\bar\rho_i$:
\begin{equation}
\bar\rho_i = \frac{1}{\tilde{A_i}} \sum_j m_j \tilde{A_j} W(\vec{x}_{ij}, h_i),
\label{eq:sph:pe:rho_bar}
\end{equation}
which can then be used to construct an entropy-weighted sound-speed
and pressure based on our assumed equation of state:
\begin{align}
\bar c_i &= c_{\rm eos}(\bar\rho_i, A_i), \label{eq:sph:pe:c_bar}\\
\bar P_i &= P_{\rm eos}(\bar\rho_i, A_i), \label{eq:sph:pe:P_bar}
\end{align}
and estimate the derivative of this later quantity with respect to the
smoothing length using:
\begin{equation}
\bar P_{\partial h_i} \equiv \dd{\bar{P}}{h}(\vec{x}_i) = \sum_j m_j
\tilde{A_j} \dd{W}{h}(\vec{x}_{ij}), \label{eq:sph:pe:P_dh}
\end{equation}
The gradient terms (``h-terms'') are then obtained by combining $\bar
P_{\partial h_i}$ 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}.
\end{equation}
\subsubsection{Hydrodynamical accelerations (\nth{2} neighbour loop)}
\subsubsection{Particle properties prediction}
\subsection{Pressure-Energy SPH}
\label{sec:sph:pu}
\cite{Hopkins2013}\\
to be done
Section 2.2.2 of \cite{Hopkins2013}.\\ \tbd
\subsection{Anarchy SPH}
Dalla Vecchia (\textit{in prep.}), also described in section 2.2.2 of
\cite{Schaller2015}.\\
\label{sec:sph:anarchy}
to be done
\tbd
\bibliographystyle{mnras}
\bibliography{./bibliography.bib}
......
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