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SWIFT
SWIFTsim
Commits
bd71c9a1
Commit
bd71c9a1
authored
Oct 10, 2016
by
Matthieu Schaller
Browse files
Finished description of PE-SPH in the documentation
parent
57f7680e
Changes
1
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theory/SPH/Flavours/sph_flavours.tex
View file @
bd71c9a1
...
...
@@ -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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