Commit d7a1830c authored by Josh Borrow's avatar Josh Borrow
Browse files

Added new information to sph_flavours about P-U SPH

parent ee8c0c1f
#!/bin/bash
python plotSoundspeed.py
pdflatex -jobname=sph_flavours sph_flavours_standalone.tex
bibtex sph_flavours.aux
pdflatex -jobname=sph_flavours sph_flavours_standalone.tex
......
......@@ -6,7 +6,8 @@ vector quantity $\Wij \equiv \frac{1}{2}\left(W(\vec{x}_{ij}, h_i) +
\nabla_x W(\vec{x}_{ij},h_j)\right)$.
%#######################################################################################################
%##############################################################################
\subsection{\MinimalSPH}
\label{sec:sph:minimal}
......@@ -33,7 +34,8 @@ and the derivative of its density with respect to $h$:
\begin{equation}
\label{eq:sph:minimal:rho_dh}
\rho_{\partial h_i} \equiv \dd{\rho}{h}(\vec{x}_i) = \sum_j m_j \dd{W}{h}(\vec{x}_{ij}
\rho_{\partial h_i} \equiv \dd{\rho}{h}(\vec{x}_i) = \sum_j m_j
\dd{W}{h}(\vec{x}_{ij}
, h_i).
\end{equation}
This corresponds to $x_i = \tilde{x}_i = m_i$, and $y_i =\tilde{y}_i = \rho_i$
......@@ -56,8 +58,8 @@ $\rho_i$ and $u_i$:
\subsubsection{Hydrodynamical accelerations (\nth{2} neighbour loop)}
We can then proceed with the second loop over
neighbours. The signal velocity $v_{{\rm sig},ij}$ between two particles is given by
We can then proceed with the second loop over neighbours. The signal velocity
$v_{{\rm sig},ij}$ between two particles is given by
\begin{align}
\mu_{ij} &=
......@@ -91,7 +93,7 @@ and the change in internal energy,
&+\left. \frac{1}{2}\nu_{ij}\vec{v}_{ij}\cdot\Big. \Wij\right], \nonumber
\end{align}
where in both cases the first line corresponds to the standard SPH
term and the second line to the viscuous accelerations.
term and the second line to the viscous accelerations.
We also compute an estimator of the change in smoothing length to be
used in the prediction step. This is an estimate of the local
......@@ -119,13 +121,17 @@ For each particle, we compute a time-step given by the CFL condition:
\Delta t = 2 C_{\rm CFL} \frac{H_i}{v_{{\rm sig},i}},
\label{eq:sph:minimal:dt}
\end{equation}
where $C_{\rm CFL}$ is a free dimensionless parameter and $H_i = \gamma h_i$ is the
kernel support size. Particles can then be ``kicked'' forward in time:
where $C_{\rm CFL}$ is a free dimensionless parameter and $H_i = \gamma h_i$ is
the kernel support size. Particles can then be ``kicked'' forward in time:
\begin{align}
\vec{v}_i &\rightarrow \vec{v}_i + \frac{d\vec{v}_i}{dt} \Delta t \label{eq:sph:minimal:kick_v}\\
u_i &\rightarrow u_i + \frac{du_i}{dt} \Delta t \label{eq:sph:minimal:kick_u}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, u_i\right) \label{eq:sph:minimal:kick_P}, \\
c_i &\rightarrow c_{\rm eos}\left(\rho_i, u_i\right) \label{eq:sph:minimal:kick_c},
\vec{v}_i &\rightarrow \vec{v}_i + \frac{d\vec{v}_i}{dt} \Delta t
\label{eq:sph:minimal:kick_v}\\
u_i &\rightarrow u_i + \frac{du_i}{dt} \Delta t
\label{eq:sph:minimal:kick_u}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, u_i\right)
\label{eq:sph:minimal:kick_P}, \\
c_i &\rightarrow c_{\rm eos}\left(\rho_i, u_i\right)
\label{eq:sph:minimal:kick_c},
\end{align}
where we used the pre-defined equation of state to compute the new
value of the pressure and sound-speed.
......@@ -136,12 +142,14 @@ 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} \\
\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}
\Delta t\right), \label{eq:sph:minimal:drift_rho} \\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, u_i + \frac{du_i}{dt} \Delta t\right), \label{eq:sph:minimal:drift_P}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, u_i + \frac{du_i}{dt} \Delta
t\right), \label{eq:sph:minimal:drift_P}\\
c_i &\rightarrow c_{\rm eos}\left(\rho_i, u_i + \frac{du_i}{dt}
\Delta t\right) \label{eq:sph:minimal:drift_c},
\end{align}
......@@ -149,7 +157,7 @@ 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}
......@@ -158,15 +166,16 @@ This flavour of SPH is the one implemented in the \gadget-2 code
\citep{Springel2005}. The basic equations were derived by
\cite{Springel2002} and also includes a \cite{Balsara1995} switch for
the suppression of viscosity. The implementation here follows closely the
presentation of \cite{Springel2005}. Specifically, we use their equations (5), (7),
(8), (9), (10), (13), (14) and (17). We summarize them here for completeness.
presentation of \cite{Springel2005}. Specifically, we use their equations (5),
(7), (8), (9), (10), (13), (14) and (17). We summarise them here for
completeness.
\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 $h$ and the ``h-terms'' in a similar way to the minimal-SPH case
smoothing length $h_i$, we compute the density, derivative of the density with
respect 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 pre-defined equation of state:
......@@ -175,13 +184,16 @@ be computed using the pre-defined equation of state:
P_i &= P_{\rm eos}(\rho_i, A_i), \label{eq:sph:gadget2:P}\\
c_i &= c_{\rm eos}(\rho_i, A_i). \label{eq:sph:gadget2:c}
\end{align}
We additionally compute the divergence and
curl of the velocity field using standard SPH expressions:
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
\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\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
\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 $B_i$:
......@@ -190,7 +202,8 @@ These are used to construct the \cite{Balsara1995} switch $B_i$:
B_i = \frac{|\nabla\cdot\vec{v}_i|}{|\nabla\cdot\vec{v}_i| +
|\nabla\times\vec{v}_i| + 10^{-4}c_i / h_i}, \label{eq:sph:gadget2:balsara}
\end{equation}
where the last term in the denominator is added to prevent numerical instabilities.
where the last term in the denominator is added to prevent numerical
instabilities.
\subsubsection{Hydrodynamical accelerations (\nth{2} neighbour loop)}
......@@ -199,7 +212,8 @@ case with the exception of the viscosity term which is now modified to
include the switch. Instead of Eq. \ref{eq:sph:minimal:nu_ij}, we get:
\begin{equation}
\nu_{ij} = -\frac{1}{2}\frac{\alpha \bar B_{ij} \mu_{ij} v_{{\rm sig},ij}}{\bar\rho_{ij}},
\nu_{ij} = -\frac{1}{2}\frac{\alpha \bar B_{ij} \mu_{ij} v_{{\rm
sig},ij}}{\bar\rho_{ij}},
\label{eq:sph:gadget2:nu_ij}
\end{equation}
whilst equations \ref{eq:sph:minimal:v_sig},
......@@ -207,8 +221,8 @@ 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. Instead of
eq. \ref{eq:sph:minimal:du_dt}, we have
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} A_{\rm eos}\left(\rho_i, \sum_j
......@@ -228,10 +242,14 @@ 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:gadget2:kick_v}\\
A_i &\rightarrow A_i + \frac{dA_i}{dt} \Delta t \label{eq:sph:gadget2:kick_A}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, A_i\right) \label{eq:sph:gadget2:kick_P}, \\
c_i &\rightarrow c_{\rm eos}\left(\rho_i, A_i\right) \label{eq:sph:gadget2:kick_c},
\vec{v}_i &\rightarrow \vec{v}_i + \frac{d\vec{v}_i}{dt} \Delta t
\label{eq:sph:gadget2:kick_v}\\
A_i &\rightarrow A_i + \frac{dA_i}{dt} \Delta t
\label{eq:sph:gadget2:kick_A}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, A_i\right)
\label{eq:sph:gadget2:kick_P}, \\
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
thermodynamical quantities.
......@@ -242,12 +260,14 @@ 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} \\
\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}
\Delta t\right), \label{eq:sph:gadget2:drift_rho} \\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, A_i + \frac{dA_i}{dt} \Delta t\right), \label{eq:sph:gadget2:drift_P}\\
P_i &\rightarrow P_{\rm eos}\left(\rho_i, A_i + \frac{dA_i}{dt} \Delta
t\right), \label{eq:sph:gadget2:drift_P}\\
c_i &\rightarrow c_{\rm eos}\left(\rho_i, A_i + \frac{dA_i}{dt}
\Delta t\right) \label{eq:sph:gadget2:drift_c},
\end{align}
......@@ -257,34 +277,37 @@ itself is \emph{not} updated.
\subsection{Weighted-Pressure SPH Validity}
A new class of Lagrangian SPH methods were introduced to the astrophysical community
by \citet{Hopkins2013} and \citet{Saitoh2013}. Two of these methods, Pressure-Entropy
(used in the original ANARCHY implementation in EAGLE) and Pressure-Energy, are
implemented for use in \swift{}. Before considering the use of these methods, though,
it is important to pause for a moment and consider where it is valid to use them
in a cosmological context. These methods (as implemented in \swift{}) are only valid
for cases that use an \emph{ideal gas equation of state}, i.e. one in which
A new class of Lagrangian SPH methods were introduced to the astrophysical
community by \citet{Hopkins2013} and \citet{Saitoh2013}. Two of these methods,
Pressure-Entropy (used in the original ANARCHY implementation in EAGLE) and
Pressure-Energy, are implemented for use in \swift{}. Before considering the
use of these methods, though, it is important to pause for a moment and
consider where it is valid to use them in a cosmological context. These methods
(as implemented in \swift{}) are only valid for cases that use an \emph{ideal
gas equation of state}, i.e. one in which
\begin{equation}
P = (\gamma - 1) u \rho \propto \rho.
\nonumber
\end{equation}
Implementations that differ from this, such as the original ANARCHY scheme in EAGLE,
may have some problems with energy conservation \cite[see][]{Hosono2013} and other
properties as at their core they assume that pressure is proportional to the local
energy density, i.e. $P \propto \rho u$. This is most easily shown in Pressure-Energy
SPH where the weighted pressure $\bar{P}$ is written as
Implementations that differ from this, such as the original ANARCHY scheme in
EAGLE, may have some problems with energy conservation \cite[see][]{Hosono2013}
and other properties as at their core they assume that pressure is proportional
to the local energy density, i.e. $P \propto \rho u$. This is most easily shown
in Pressure-Energy SPH where the weighted pressure $\bar{P}$ is written as
\begin{equation}
\bar{P} = \sum_j \frac{P_j}{\rho_j} W_{ij} = \sum_j m_j (\gamma - 1) u_j W_{ij},
\bar{P} = \sum_j \frac{P_j}{\rho_j} W_{ij} = \sum_j m_j (\gamma - 1) u_j
W_{ij},
\end{equation}
and the right-hand side is what is actually calculated using the scheme. It is
clear that this does not give a valid weighted pressure for any scheme using a non-ideal
equation of state. Fortunately, there is a general prescription for including non-ideal
equations of state in the P-X formalisms, but this is yet to be implemented in \swift{}
and requires an extra density loop. Attempting to use these weighted schemes with
a non-ideal equation of state will lead to an incorrect calculation of both the pressure
and the equation fo motion. How incorrect this estimate is, however, remains to be seen.
clear that this does not give a valid weighted pressure for any scheme using a
non-ideal equation of state. Fortunately, there is a general prescription for
including non-ideal equations of state in the P-X formalisms, but this is yet
to be implemented in \swift{} and requires an extra density loop. Attempting to
use these weighted schemes with a non-ideal equation of state will lead to an
incorrect calculation of both the pressure and the equation fo motion. How
incorrect this estimate is, however, remains to be seen.
%#####################################################################################################
%##############################################################################
\subsection{Pressure-Entropy SPH}
\label{sec:sph:pe}
......@@ -312,7 +335,8 @@ 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),
\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
......@@ -346,8 +370,11 @@ P_{\partial h_i}$ and $\rho_{\partial h_i}$ (eq. \ref{eq:sph:minimal:rho_dh}):
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) \\
\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
......@@ -370,9 +397,11 @@ 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}\\
\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}, \\
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}
......@@ -387,14 +416,16 @@ 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} \\
\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}\\
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}
......@@ -428,7 +459,8 @@ with the resulting variables representing a smoothed pressure and particle
number density. This choice of variables leads to the following equation of
motion:
\begin{align}
\frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t} = -\sum_j (\gamma - 1)^2 m_j u_j u_i
\frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t} = -\sum_j (\gamma - 1)^2 m_j u_j
u_i
&\left[\frac{f_{ij}}{\bar{P}_i} \nabla_i W_{ij}(h_i) ~+ \right. \nonumber \\
&\frac{f_{ji}}{\bar{P}_j} \nabla_i W_{ji}(h_j) ~+ \nonumber \\
& \left.\nu_{ij}\bar{\nabla_i W_{ij}}\right]~.
......@@ -439,7 +471,8 @@ the final term.
The $h$-terms are given as
\begin{align}
f_{ij} = 1 - \left[\frac{h_i}{n_d (\gamma - 1) \bar{n}_i \left\{m_j u_j\right\}}
f_{ij} = 1 - \left[\frac{h_i}{n_d (\gamma - 1) \bar{n}_i \left\{m_j
u_j\right\}}
\frac{\partial \bar{P}_i}{\partial h_i} \right]
\left( 1 + \frac{h_i}{n_d \bar{n}_i}
\frac{\partial \bar{n}_i}{\partial h_i} \right)^{-1}
......@@ -458,45 +491,104 @@ evolved. Following \cite{Hopkins2013}, this is calculated as
\nabla_i W_{ij}(h_i)~.
\label{eq:sph:pu:dudt}
\end{align}
The sound-speed in P-U requires some consideration. To see what the `correct' sound-speed
is, it is worth looking at the equation of motion (Equation \ref{eq:sph:pu:eom}) in
contrast with that of the EoM for Density-Energy SPH (Equation \ref{eq:sph:minimal:dv_dt})
to see what terms are applicable.
For Density-Energy SPH, we see that
The sound-speed in P-U requires some consideration. To see what the `correct'
sound-speed
is, it is worth looking at the equation of motion (Equation
\ref{eq:sph:pu:eom}) in
contrast with that of the EoM for Density-Energy SPH (Equation
\ref{eq:sph:minimal:dv_dt})
to see what terms are applicable. For Density-Energy SPH, we see that
\begin{align}
\frac{\mathrm{d}\mathbf{v}_i}{\mathrm{d} t} \sim \frac{c_{s, i}}{\rho_i} \nabla_i W_{ij},
\frac{\mathrm{d}\mathbf{v}_i}{\mathrm{d} t} \sim \frac{c_{s, i}}{\rho_i}
\nabla_i W_{ij},
\nonumber
\end{align}
and for Pressure-Energy SPH
\begin{align}
\frac{\mathrm{d}\mathbf{v}_i}{\mathrm{d} t} \sim (\gamma - 1)^2 \frac{u_i u_j}{\bar{P}_i}
\nabla_i W_{ij}.
\frac{\mathrm{d}\mathbf{v}_i}{\mathrm{d} t} \sim (\gamma - 1)^2
\frac{u_i u_j}{\bar{P}_i} \nabla_i W_{ij}.
\nonumber
\end{align}
From this it is reasonable to assume that the sound-speed, i.e. the speed at which
information propagates in the system through pressure waves, is given by the expression
From this it is reasonable to assume that the sound-speed, i.e. the speed at
which information propagates in the system through pressure waves, is given by
the expression
\begin{align}
c_{s, i} = (\gamma - 1) u_i \sqrt{\gamma \frac{\rho_i}{\bar{P_i}}}.
\label{eq:sph:pu:soundspeedfromeom}
\end{align}
This expression is dimensionally consistent with a sound-speed, and includes
the gas density information (through $\rho$), traditionally used for
sound-speeds, as well as including the extra information from the smoothed
pressure $\bar{P}$. However, this scheme causes some problems, which can be
illustrated using the Sedov Blast test. Such a sound-speed leads to a
considerably \emph{higher} time-step in front of the shock wave (where the
smoothed pressure is higher, but the SPH density is relatively constant),
leading to integration problems. An alternative to this is to use the smoothed
pressure in the place of the ``real" pressure. Whilst it is well understood
that $\bar{P}$ should not be used to replace the real pressure in general, here
(in the sound-speed) it is only used as part of the time-stepping condition.
Using
\begin{align}
c_{s, i} = \sqrt{\gamma \frac{\bar{P}_i}{\rho_i}}
\label{eq:sph:pu:soundspeed}
\end{align}
This expression is dimensionally consistent with a sound-speed, and includes the gas
density information (through $\rho$), traditionally used for sound-speeds, \emph{and}
the information from the equation of motion (through $\bar{P}$). This sound-speed is
used in the artificial viscosity scheme and in the calculation of the signal velocity,
as was shown in the section for \MinimalSPH.
instead of Equation \ref{eq:sph:pu:soundspeedfromeom} leads to a much improved
time-stepping condition that actually allows particles to be woken up before
being hit by a shock (see Figure \ref{fig:sph:soundspeed}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{sedov_blast_soundspeed.pdf}
\caption{The ratio of the sound-speed calculated in Equation
\ref{eq:sph:pu:soundspeed} to the gas sound-speed,
$c_s = \sqrt{\gamma(\gamma - 1) u}$ with $u$ the internal energy. Note how
the sound-speed increases ahead of the shock, leading to a much smaller
time-step for these particles ($\Delta t \propto c_s^{-1}$), waking them up
just before they are hit by a shock. The physical reasoning behind using
this particular metric for the sound-speed is weak, but as a time-step
criterion it appears to be useful. The smoothed pressure calculation
``catches" the hot particles from the shock that is incoming and is
increased for those in front of the shock. Thankfully, particles that are
behind the shock seem to be relatively unaffected. The red line shows the
shock front.}
\label{fig:sph:soundspeed}
\end{figure}
\subsubsection{Time integration}
Time integration follows exactly the same scheme as \MinimalSPH.
Time integration follows exactly the same scheme as \MinimalSPH, as the
fundamental equations are exactly the same (just slightly different quantities
enter the equations of motion). The CFL condition is used, along with the
sound-speed that is discussed above, such that
\begin{align}
\Delta t = 2 C_{\rm CFL} \frac{H_i}{v_{{\rm sig},i}},
\label{eq:sph:pu:dt}
\end{align}
where $C_{\rm CFL}$ is a free dimensionless parameter and $H_i = \gamma h_i$ is
the kernel support size. There is an additional requirement placed on the
maximal change in $u$ such that
\begin{align}
\Delta t = C_{u} \frac{u}{du/dt},
\label{eq:sph:pu:dt_du}
\end{align}
with $C_{u}$ a free dimensionless parameter that describes the maximal change
in $u$ that is allowed.
\subsubsection{Particle properties prediction}
The prediciton of particle properties follows exactly the same scheme as
\MinimalSPH. \TODO: P-U also includes some efforts to drift the smoothed pressure.
It is unclear at the moment what form that should take.
The prediction of particle properties follows exactly the same scheme as
\MinimalSPH, with the exception of course of the additional smoothed quantity
$\bar{P}$. This is drifted in the same way as the density; they should both
follow from the same differential equation with an additional $u$ factor on
both sides, such that
\begin{align}
\bar{P}_i \rightarrow \bar{P}_i
\exp\left(-\frac{3}{h_i}\frac{dh_i}{dt} \Delta t\right).
\label{eq:sph:pu:drift}
\end{align}
%##############################################################################
\subsection{Anarchy SPH}
Dalla Vecchia (\textit{in prep.}), also described in section 2.2.2 of
......
......@@ -4,6 +4,10 @@ python kernels.py
cp kernels.pdf ..
cp kernel_derivatives.pdf ..
cd ..
cd Flavours
python plotSoundspeed.py
cp sedov_blast_soundspeed.pdf ..
cd ..
pdflatex swift_sph.tex
bibtex swift_sph.aux
pdflatex swift_sph.tex
......
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