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SWIFT
SWIFTsim
Commits
fe92c7d0
Commit
fe92c7d0
authored
May 09, 2020
by
Matthieu Schaller
Browse files
Complete the discussion of the truncated MAC estimator
parent
bfa215a1
Changes
1
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theory/Multipoles/fmm_mac.tex
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fe92c7d0
...
...
@@ 189,7 +189,25 @@ f_{\rm MAC}(r) =
\right
.
\label
{
eq:fmm:f
_
mac
}
\end{align}
This esimator is shown as a dotdashed line on
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
)
$
the true
truncated and softened forces (green line).
Since it is made of constants and even powers of the distance,
computin this term is much cheaper than the true forces. This
esimator is shown as a dotdashed line on
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 nontruncated unsoftened 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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