Biology Reference
In-Depth Information
i
i
i
Ú
xxx
h
(
xx
d
=
0
if
i
+
i
+
i
=
1
or
3
£
i
+
i
+
i
£
r
+
1
123
123
123
1
23
3
IR
(23)
r
+
2
Ú
x
h
()d
x
x
<•
(24)
2
3
IR
for
i
1
,
i
2
,
i
3
∈
j
and 0 otherwise. The first condition
normalizes the kernel function. The second one requires all moments up
to order
r
IN
0
, and
δ
ij
=
1 if
i
=
1 to vanish, and the third one is required in order to bound
the truncation error. Using the requirement in Eq. (21), the only
remaining terms in Eq. (20) are
+
-
2
2
r
Ú
(()
uu
y
-
()) (
x
h
x
-
y
)
d
y
= —
u
()
x
+
O
( ),
(25)
d
IR
and the integral operator that approximates the Laplacian is found as
2
-
2
Ú
—=
u
()
x
(()
u
y
-
u
()) (
x
h
x
-
y
) .
d
y
d
(26)
IR
While this operator is not the only possibility of discretizing the
Laplacian onto particles, it has the big advantage of conserving
mass exactly.
85
(
r
), with
r
being the
largest integer for which the condition in Eq. (21) is fulfilled.
85
Equation (26) is discretized using the particle locations as quadrature
points. Thus,
The approximation error is
O
N
Â
2
h
h
-
2
h
h
h
h
—
u
()
x
=
(
V u
-
V u
)(
h
x
-
x
,
(27)
,
h
p
qq
qp
p
q
q
=
1
where
V
q
is the volume of particle
q
. Inserting this discretized operator
into Eq. (15), the final PSE scheme for isotropic diffusion reads
