T. Sakuma
310
( )
( )
1
,.
N
i j ij
j
Fx C Kxz
=
= ⋅
∑
(1)
However, naive computation of the direct summation results in an
( )
2
ON
computational cost. The Cartesian treecode [13] is a type of treecode used to
evaluate the screened Coulomb (Yukawa) potential efficiently, which involves
potential
( )
,
ij
Kxz
given by
( )
e
,: .
ji
zx
ij
ji
Kxz
zx
κ
−−
=
−
(2)
The intuitive idea behind treecode is that one imagines wanting to compute
gravitational force
( )
i
Fx
of a star located at some point
i
x
. Given clusters of
stars which are very far from the target star, instead of summing the gravita-
tional force of all the stars in the clusters, if
D
/
r
is very small, where
D
is the size
of the cluster and r is the distance of the cluster point from the target point, then
we can approximate the force for each cluster as that of a star whose mass is
equal to the total sum of the stars in the cluster and is located at the center of the
cluster (called the cluster point). Treecode applies the Barnes-Hut algorithm [18]
to build a hierarchy of such clusters and employs an efficient recursive compu-
tation of (far-field) Taylor coefficients to compute particle-cluster approxima-
tions of the screened Coulomb potential, reducing the computational cost from
( )
2
ON
to
(
)
log
ON N
.
A more concrete explanation of the Cartesian treecode is as follows (see [13]
for details). First, particles
( )
12
,,,
n
xx x
are divided into a hierarchy of clus-
ters c’s by the Barnes-Hut algorithm,
( ) ( ) ( )
( )
, ,.
j
i ci ci i ij
c zc
Fx Fx Fx CKxz
∈
= = ⋅
∑∑
(3)
Second, truncated expansion of the Taylor series of
( )
,
ij
Kxz
up to order
p
around cluster point
c
z
gives
( )
( )
( )
0
,
1
,.
!
c
k
p
k
i
ij j c
k
k
zz
Kxz
Kxz z z
k
z
=
=
∂
= −
∂
∑
(4)
For each cluster
c
, the Taylor expansion is applied if
r
R
θ
≤
, where
r
is the
radius of the cluster,
R
is the distance from
i
x
to the cluster
c
, and
θ
is the
multipole acceptance criterion (MAC) parameter. If the condition is not satisfied,
then the children of c are examined, or simple direct summation is applied if the
cluster is a leaf of the tree. Finally, substituting (4) into (3) gives
( )
(
)
( )
( )
( )
(
)
0
,
00
,
1
!
,
1
: ,.
!
j
c
j
c
k
p
k
i
ci j j c
k
zc k
zz
k
pp
k
i
j j c k i c ck
k
k zc k
zz
Kxz
Fx C z z
k
z
Kxz
C z z L xz M
k
z
∈=
=
= ∈=
=
∂
=⋅−
∂
∂
= −=
∂
∑∑
∑ ∑∑
(5)
Because
( )
,
k ic
L xz
is independent of
j
z
and
,ck
M
is independent of
i
x
,
( )
,
k ic
L xz
and
,ck
M
can be calculated independently and computation of
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