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 ) .
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