Biomedical Engineering Reference
In-Depth Information
Here, the finite difference operators
D
±
x
are defined by
D
+
x
φ
i
,
j
=
φ
i
+
1
,
j
−
φ
i
,
j
x
D
−
x
φ
i
,
j
=
φ
i
,
j
−
φ
i
−
1
,
j
x
(4.10)
,
.
The operators
D
±
y
are defined in a similar manner for the
j
th index. Note that
the numerical flux function, the term multiplied by
t
in Eq. 4.9, senses the
direction in which the interface is moving, then chooses the finite difference
approximation which looks in the correct direction, also known as the upwind
direction.
A second-order method based upon the ENO method [55] is given by
n
+
1
ij
ij
−
t
(max( sign(
F
ij
)
A
,
−
sign(
F
ij
)
B
,
0)
2
+
max( sign(
F
ij
)
C
,
−
sign(
F
ij
)
D
,
0)
2
)
1
/
2
n
φ
=
φ
,
(4.11)
where
+
x
2
n
ij
n
ij
n
A
=
D
−
x
φ
minmod(
D
−
x
D
−
x
φ
,
D
−
x
D
+
x
φ
ij
)
,
(4.12)
ij
+
x
n
n
n
B
=
D
+
x
φ
minmod(
D
+
x
D
−
x
φ
ij
,
D
+
x
D
+
x
φ
ij
)
,
(4.13)
2
+
y
2
ij
ij
ij
)
,
C
=
D
−
y
φ
minmod(
D
−
y
D
−
y
φ
,
D
−
y
D
+
y
φ
(4.14)
ij
+
y
2
n
n
n
D
=
D
+
y
φ
minmod(
D
+
y
D
−
y
φ
ij
,
D
+
y
D
+
y
φ
ij
)
,
(4.15)
and where
1
2
( sign(
a
)
+
sign(
b
)) min(
|
a
|
,
|
b
|
)
.
minmod(
a
,
b
)
=
(4.16)
In general, the speed function,
F
, in Eq. 4.5 is split into
F
=
F
adv
+
F
diff
, where
F
adv
is the advective part and
F
diff
is the diffusive part. When constructing the
numerical method for solving Eq. 4.5, the numerical flux function is used for
the advective part, and the diffusive part is discretized using standard central
differences.
To illustrate this, we take an example used in [85], where
F
=
1
−
κ
,0
<
<<
1, and
κ
is the mean curvature given by
2
y
+
φ
yy
φ
2
x
−
2
φ
xy
φ
x
φ
y
κ
=
φ
xx
φ
(4.17)
.
φ
y
3
/
2
x
+
φ
In this example,
F
is broken down so that
F
adv
=
1 and
F
diff
=−
κ
. Using Go-
dunov's method for the advective term and central differences for the diffusive
