Biology Reference
In-Depth Information
using Green's function
55
G
(
x
,
y
).
b
The resulting integral defines an exten-
sive quantity that is then discretized and computed by a quadrature
84
with some weights
w
. The values of the weights depend on the particu-
lar quadrature rule used. For midpoint quadrature
102
and the example of
Fig. 4, they would be
w
q
f
(
y
q
)d
y
. This defines the right-hand side of
Eq. (1) for this example. The advantages of the latter procedure are that
the integral solution is always consistent (even analytically exact), and
that numerical quadrature is always stable. The only property that
remains to be concerned about is the solution's accuracy. The first way
of solution is sometimes referred to as the “intensive method”, and the
second as the “extensive method”.
=
5.1. Function Approximation by Particles
The approximation of a continuous field function
u
(
x
) by particles in
d
-dimensional space can be developed in three steps:
•
Step 1: integral representation
. Using the Dirac
-function identity,
the function
u
can be expressed in integral form as
δ
Ú
u
()
x
=
u
() (
y
d
x
-
y
) .
d
y
(2)
In point particle methods such as random walk (cf. Sec. 7.1), this
integral is directly discretized on the set of particles using a
quadrature rule with the particle locations as quadrature points.
Such a discretization, however, does not allow recovering the
function values at locations other than those occupied by the
particles.
•
Step 2
:
integral mollification
. Smooth particle methods relax this
limitation by regularizing the
δ
-function by a mollification kernel
ζ
=
−d
ζ
(
x
/
), with lim
→0
ζ
= δ
, that conserves the first
r
−
1
moments of the
ζ
can be thought
of as a cloud or blob of strength, centered at the particle location,
δ
-function identity.
85
The kernel
b
Note that Green's function always exists, even though it may not be known in
closed form in most cases.
