Biomedical Engineering Reference
In-Depth Information
ing, repositioning, and connecting markers is a tedious task, typically for higher-
dimensional cases. Our aim in this section is to present numerical schemes for
interface evolution problems. This presentation will be based on the link between
Hamilton-Jacobi HJ equations and hyperbolic conservation laws (HCLs).
Although these two views are formally equivalent for only one dimensional
case, multidimensional schemes are motivated using the 1D numerical methodol-
ogy. We shall borrow the schemes designed for HCLs to motivate our schemes for
HJ equations.
4.2.6. Hamilton-Jacobi equations
Definition
An equation of the form
U
t
+
H
(
x, y, z, U
x
,U
y
,U
z
)=0
(9)
is called a Hamilton-Jacobi equation. The function
H
(
·
) is called the Hamiltonian.
Note that our level set equation (7) can be cast in the form of an HJ equation with
H
(
x, y, z, p, q, r
)=
F
p
2
+
q
2
+
r
2
.
Let us focus on a one-dimensional version, that is,
U
t
+
H
(
U
x
)=0. Ifwelet
u
=
U
x
and differentiate w.r.t.
t
, we get the following:
d
dt
(
U
x
)=
d
dx
(
U
t
)=
d
dx
(
−
d
dx
(
−
u
t
=
H
(
U
x
)) =
H
(
u
)) =
−
[
H
(
u
)]
x
⇒
u
t
+[
H
(
u
)]
x
=0
,
(10)
which is anHCL. The corresponding flux function is equal to theHamiltonian
H
(
·
).
(Given an HCL
u
t
+[
G
(
u
)]
x
=0and a discretized grid in space
{
···
,x
n
}
x
0
,x
1
,
,
.
=
a “numerical flux” function is a function
g
such that
g
(
u
i−
1
,u
i
)
G
(
u
i−
2
)
G
i−
2
, that is,
g
approximates the flux
G
at the half grid points.) Thus, in order
to solve our Hamilton-Jacobi equation, we first formulate it as an HCL, borrow
numerical flux schemes designed for an HCL [37], and then return back to the
original equation.
Given that
u
=
U
x
, one can notice that the forward difference approximation
D
+
x
U
i
U
i
+1
−
U
i
∆
x
=
to
u
i
is actually the central difference approximation to
U
i
−
U
i
−
1
∆
x
u
i
+
2
. Likewise, the backward approximation
D
−x
U
i
=
is the central
difference approximation to
u
i−
2
. Moreover, from the definition of the numerical
function, one can deduce that
g
(
u
i−
2
,u
i
+
2
)
H
(
u
i
)
.
Hence, the equation
U
t
+
H
(
u
)=0yields the following scheme:
U
n
+1
i
=
U
i
−
∆
tg
(
D
−x
U
i
,D
+
x
U
i
)
.