Biology Reference
In-Depth Information
In the framework of pure particle methods, continuous diffusion
models can be simulated using particles carrying mass as their extensive
strength ω and collectively representing the intensive concentration field u .
In the following, we review the stochastic method of random walk (RW)
and the deterministic particle strength exchange (PSE) method. Using a
1D test problem, we then compare the accuracy and the convergence
behavior of the two methods.
7.1. The Method of Random Walk (RW)
The Random Walk (RW) 116,120 method is based on the stochastic inter-
pretation of Green's function solution 55
(cf. Fig. 4) of the diffusion
equation:
= Ú
ut
(,)
x
G
(, ,) () .
x y
tu
y
d
y
(16)
0
In the case of d -dimensional isotropic homogeneous free-space diffu-
sion, i.e. D
=
D
and
Ω=
IR d , Green's function is explicitly known
to be 121
2
È
˘
xy
-
1
Í
Í
2
˙
˙
.
Gt
(, ,)
xy
=
exp
-
(17)
d
2
4
Dt
(
4
p
Dt
)
Î
˚
The RW method interprets this function as the transition density of a sto-
chastic process. 122 In d dimensions, the method starts by either uniformly
or randomly placing N particles p at initial locations x p , p
=
1, 2,…, N .
Each particle is assigned a strength of ω p =
V p u 0 ( x p ), where V p is the par-
ticle volume. This defines a point particle function approximation (cf.
Sec. 5.1) to the initial concentration field u 0 ( x ). The particles then
undergo a random walk by changing their positions at each positive-
integer time step n according to the transition density in Eq. (17):
n
+
1
n
n
N
x
=+
x
(,
02
Dt
d
),
(18)
p
p
p
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