Biology Reference
In-Depth Information
forces (electrostatics). In a hybrid PM approach, such long-range
interactions are modeled by a corresponding field equation that is
then solved on the mesh. In many applications, the fields to be dis-
cretized are gradient fields, such that the corresponding long-range
operator is the Laplace operator and the field equation hence is the
Poisson equation (cf. Fig. 4). This equation can efficiently be solved
using, e.g. FD
79
implemented in a multigrid algorithm,
108
or Poisson
solvers based on fast Fourier transforms. In hybrid PM methods, the
functions
K
and
F
in Eq. (1) may thus contain contributions corre-
sponding to the solution of the field equation on the mesh.
Therefore, hybrid methods require
•
interpolation of the
ω
p
carried by the particles from the irregular
particle locations
x
p
onto the
M
regular mesh points (
ω
m
):
N
∑
h
h
h
ω
=
Q
(
xx
−
)
ω
m
=
12
, ,
K
,
M
;
(10)
m
mp
p
p
1
•
and interpolation of the field solution
F
m
(and, similarly,
K
m
if present) from the mesh to the (not necessarily same) particle
locations (
F
p
):
M
∑
h
h
h
F
=
R
(
x
−
x
)
F
p
=
12
, ,
K
,
N
.
(11)
p
p
mm
m
1
The accuracy of the method depends on the smoothness of
K
and
F
,
on the interpolation functions
Q
and
R
, and on the mesh-based discretiza-
tion scheme employed for the solution of the field equations. In order to
achieve high accuracy, the interpolation functions
Q
and
R
must be
smooth to minimize local errors, and conserve the moments of the inter-
polated quantity to minimize far-field errors.
27
In addition, it is necessary
that
Q
is at least of the same order of accuracy as
R
in order to avoid spu-
rious contributions to
F
p
.
107
This can easily be achieved by selecting the
same interpolation function,
W
, for both operations:
Q
W
. Accurate
interpolation functions that conserve the moments of the interpolated
=
R
=
