Biology Reference
In-Depth Information
h
d
d
d
d
x
p
=
0
t
h
(28)
ω
N
p
Â
-
2
h
h
h
h
=
V D
(
Vu
-
Vu
)
h
(
xx
-
)
p
q
q
qp
p
q
t
q
=
1
p
=
12
,, ,
K
N
.
η
is local and the fast algorithms described in Sec. 6.1 can
be used to reduce the computational complexity to
The PSE kernel
(
N
). In order to
simulate diffusion using PSE, the strengths of all the particles change
(i.e. they exchange mass) while their locations remain constant (i.e. they
do not move). This is dual to the method of RW, where the
particles conserve their mass but move in space. In PSE, all geometry and
boundary condition processing thus only needs to be done once when
initializing the particles. Combined convection-diffusion problems can
be simulated by moving the particles with the convective velocity field
instead of keeping them fixed.
Besides the obvious choice of using a Gaussian [cf. Eq. (17)] as the
PSE kernel
O
, various algebraic kernels have also been derived. Algebraic
kernels are computationally more efficient, since evaluating the expo-
nential function on the floating-point unit of a computer processor takes
several tens of clock cycles. In the 3D case, the following second-order
accurate kernel as proposed by G.-H. Cottet (private communication,
1999) can, for example, be used:
η
15
1
.
h
()
x
=
(29)
2 0
p
x
+
1
7.2.2. Boundary conditions in PSE
The PSE algorithm as described so far only applies to infinite domains or
to particles farther away from the boundary than
r
c
. For particles within
an
r
c
-neighborhood from the boundary, we need to modify the PSE
scheme in order to account for the prescribed boundary conditions.
