Biology Reference
In-Depth Information
For homogeneous boundary conditions in the case of flat (compared to
the core size
of the mollification kernel) boundaries, a straightforward
method consists of placing mirror particles in an
r
c
-neighborhood out-
side of the simulation domain (Fig. 7). In the resulting method of
images, the integral operator becomes
-
2
r
Ú
O
( ( )
uu
y
-
( ))(
x
h
(
xy
-
)
±
h
(
xy y
+
))
d
+
(
).
(30)
d
IR
The final scheme is thus represented as
ω
h
d
d
N
-
2
Â
h
h
h
h
h
h
=
V D
(
Vu
-
Vu
)(
h
(
xx
-
)
±
h
(
xx
+
)).
(31)
p
q
q
qp
p
q
p
q
t
q
=
1
The positive sign between the two kernel functions applies for homoge-
neous Neumann boundary conditions, whereas the negative sign is to
be used in the case of homogeneous Dirichlet boundary conditions. The
method of images is restricted to the case of homogeneous boundary
conditions. For inhomogeneous boundary conditions, the particle
strengths need to be adjusted in the vicinity of the boundary.
123
7.3. Comparison of PSE and RW
The accuracy of the RW and PSE methods is illustrated by using a bench-
mark case of isotropic homogeneous diffusion on the 1D (
d
=
1) ray
∞
Ω=
[0,
), subject to the following initial and boundary conditions:
2
Ï
-
x
Ô
Ô
uxt
(,
==
0
)
u x
()
=
xe
x
Υ
[, ),
0
t
=
0
0
(32)
ux
(
==
00
, )
t
x
= <£
00
,
t
T
.
The exact analytic solution of this problem is
x
Dt
2
-
x
/(
14
+
t
)
uxt
ex
(,)
=
e
.
(33)
32
(
14
+
)
