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SWIFT
SWIFTsim
Commits
fe92c7d0
Commit
fe92c7d0
authored
5 years ago
by
Matthieu Schaller
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Complete the discussion of the truncated MAC estimator
parent
bfa215a1
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1 merge request
!1077
Improved multipole acceptance criterion (MAC)
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theory/Multipoles/fmm_mac.tex
+22
-4
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theory/Multipoles/fmm_mac.tex
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4 deletions
theory/Multipoles/fmm_mac.tex
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−
4
View file @
fe92c7d0
...
@@ -189,7 +189,25 @@ f_{\rm MAC}(r) =
...
@@ -189,7 +189,25 @@ f_{\rm MAC}(r) =
\right
.
\right
.
\label
{
eq:fmm:f
_
mac
}
\label
{
eq:fmm:f
_
mac
}
\end{align}
\end{align}
This esimator is shown as a dot-dashed line on
Since it is made of constants and even powers of the distance,
Fig.
\ref
{
fig:fmm:mac
_
potential
}
and obeys the relation
$
f
_{
\rm
SWIFT
}
(
r
)
computin this term is much cheaper than the true forces. This
\leq
f
_{
\rm
MAC
}
(
r
)
\leq
1
/
r
^
2
$
, with
$
f
_{
\rm
SWIFT
}
(
r
)
$
the true
esimator is shown as a dot-dashed line on
truncated and softened forces (green line).
Fig.
\ref
{
fig:fmm:mac
_
potential
}
and obeys the relation
$
f
_{
\rm
SWIFT
}
(
r
)
\leq
f
_{
\rm
MAC
}
(
r
)
\leq
1
/
r
^
2
$
, with
$
f
_{
\rm
SWIFT
}
(
r
)
$
being the true truncated and softened norm of the gravity forces the
code solves for (green line). We use this expression in the multipole
acceptance criterion instead of the
$
1
/
|
\mathbf
{
R
}
|
$
term:
\begin{equation}
\tilde
{
E
}_{
BA,p
}
M
_
Bf
_{
\rm
MAC
}
(|
\mathbf
{
R
}
|) <
\epsilon
_{
\rm
FMM
}
\min
_{
a
\in
A
}
\left
(|
\mathbf
{
a
}_
a|
\right
).
\label
{
eq:fmm:mac
_
f
_
mac
}
\end{equation}
The same change is applied to the MAC used of the M2P kernel
(eq.
\ref
{
eq:fmm:mac
_
m2p
}
). In the non-truncated un-softened case, this
expression reduces to
\citep
{
Dehnen2014
}
one. Using this expression
instead of the simpler Newtonian one only makes a difference in
simulations where a lot of particles cluster below the scale of the
softening, which is often the case for hydrodynamical simulations
including radiative cooling processes. The use of this term over the
simpler
$
1
/
r
^
2
$
estimator is a runtime parameter.
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