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any differential operator by a corresponding integral. 105
Following this
framework, a differential operator L β of order
applied to a continuous
function u ( x ) is equivalent to the integral operator
β
1
b
b
Ú
Lu
()
x
=
(()
u
y
±
u
()) (
x
h
x
-
y
)
d
y
(8)
||
b
with a suitable scaled kernel
η β ( x / ) of core size . This inte-
gral operator is then discretized over the particles using, e.g., midpoint
quadrature 102 of resolution h , yielding
η β ( x )
= d
1
b
Â
b
Lu
()
x
=
V u
(() ( ) (
x
±
u
x
h
x
-
x
.
(9)
p
q
q
p
p
q
h
||
b
q
Pure particle methods thus amount to evaluating direct particle-
particle (PP) interactions, which means that for each particle
p
1, 2,…, N
[cf. Eq. (1)]. The computational complexity of this N -body problem
thus nominally scales as
=
1, 2,…, N we have to compute a sum over all particles q
=
O
( N 2 ). Efficient algorithms do, however, exist to
reduce it to
( N ) in all practical cases. These algorithms will be outlined
in Sec. 6. Alternatively, hybrid PM methods, as described next, can be used.
O
5.2.2. Hybrid particle-mesh (PM) methods
In hybrid PM methods, as pioneered by Harlow, 106 some (but not all)
of the differential operators are evaluated on a superimposed regular
Cartesian mesh. 107 This amounts to splitting the operators into sepa-
rate short-range and long-range contributions. The short-range inter-
actions are directly evaluated on the particles, whereas the long-range
contributions are evaluated on the mesh. Using direct PP interactions
for the short-range part allows better resolving local phenomena and
retaining the favorable stability properties of particle methods in the
case of convection (moving the particles is a local operation).
Prominent examples of hybrid PM methods can be found in fluid
dynamics and electrostatics. Both applications involve long-range
interactions in order to compute the velocities (fluid dynamics) or
 
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