Biomedical Engineering Reference
In-Depth Information
derivatives are discretized by using “difference” and (2) solving the derived dif-
ference equation—an algebraic equation—by using either an iterative or a direct
method.
To begin with, we first partition the domain
by a mesh grid. For example,
we use a uniform mesh grid with grid lines:
x
j
=
x
0
+
jh
,
j
=
0
,
1
,...,
J
,
y
l
=
y
0
+
lk
,
l
=
0
,
1
,...,
L
,
where
h
=
x
i
+
1
−
x
i
and
k
=
y
i
+
1
−
y
i
are the mesh sizes in the
x
and
y
directions,
respectively. For simplicity, we write
f
j
,
l
=
f
(
x
j
,
y
l
), the function values on the
nodes of the mesh.
Using Taylor expansion and intermediate value theorem, we can derive the
following numerical differentiation formulas:
forward difference formula:
1
h
(
u
i
+
1
,
j
−
u
i
,
j
)
,
u
x
(
x
,
y
)
≈
(5.23)
1
k
(
u
i
,
j
+
1
−
u
i
,
j
);
u
y
(
x
,
y
)
≈
backward difference formula:
1
h
(
u
i
,
j
−
u
i
−
1
,
j
)
,
u
x
(
x
,
y
)
≈
(5.24)
1
k
(
u
i
,
j
−
u
i
,
j
+
1
);
u
y
(
x
,
y
)
≈
centered difference formula:
1
2
h
(
u
i
+
1
,
j
−
u
i
−
1
,
j
)
,
u
x
(
x
,
y
)
≈
(5.25)
1
2
k
(
u
i
,
j
+
1
−
u
i
,
j
−
1
)
.
Similarly, the three second-order partial derivatives are given by
u
y
(
x
,
y
)
≈
1
hk
(
u
i
+
1
,
j
−
2
u
i
,
j
+
u
i
−
1
,
j
)
,
u
xx
(
x
,
y
)
≈
1
hk
(
u
i
,
j
+
1
−
2
u
i
,
j
+
u
i
,
j
−
1
)
,
u
yy
(
x
,
y
)
≈
(5.26)
1
hk
(
u
i
+
1
,
j
+
1
−
2
u
i
,
j
+
u
i
−
1
,
j
−
1
)
,
Formulas (5.23-5.26) will be used in Section 5.3 to discretize (or linearize) the
irradiance equation (5.2).
u
xy
(
x
,
y
)
≈
