Biomedical Engineering Reference
In-Depth Information
with
A
A
0
m
A
A
0
m
ψ
A
=
∂ψ
∂
mK
A
,
ψ
K
=
∂ψ
∂
A
=
K
=
−
1
.
(2.9)
The complete system reads
⎫
⎬
+
∂
(
)=
,
∂
t
A
uA
0
x
(2.10)
⎭
A
ρ
ψ
A
∂
x
A
A
ρ
ψ
K
∂
x
K
Au
2
∂
t
(
uA
)+
∂
x
(
)+
=
−
−
∂
x
p
e
(
x
)
−
Ru
.
We note that the principal part of the equations (left-hand side) does not have
conservation-law form. Note also that there are source terms on the right hand side
which depend on gradients of the vessel properties
E
(
x
)
and
h
0
(
x
)
and the external
pressure. The external pressure
p
e
(
is analogous to bottom variation in shallow
water flows [21], both giving rise to a source term involving a spatial gradient. In
the rest of this paper we assume
p
e
(
x
)
to be constant. In the numerical literature it is
well known that the treatment of such source terms, sometimes known as
geometric
source terms
, is notoriously difficult. Common difficulties include the generation of
spurious oscillations and the lack of balance between convective terms and source
terms in the steady state. For a discussion on these issues see, for example, [9, 13, 16]
and references therein. In principle, for slowly-varying vessel properties one can still
proceed with formulation (2.10). However, for significant vessel property variations,
or even in the case of discontinuous properties, formulation (2.10) is not suitable.
In this paper we present an alternative formulation of the model by considering the
variable vessel properties
h
0
(
x
)
as additional unknowns of the problem. As
a matter of fact it is sufficient to consider the coefficient
K
x
)
and
E
(
x
)
(
x
)
as the new unknown, as
this includes the combined variations of
h
0
(
x
)
and
E
(
x
)
. We then add the following
obvious partial differential equation
∂
t
K
(
x
)=
0
.
(2.11)
The enlarged system from (2.10) and (2.11) in quasi-linear form reads
∂
t
Q
+
A
(
Q
)
∂
x
Q
=
S
(
Q
)
,
(2.12)
where
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
q
1
q
2
q
3
A
Au
K
s
1
s
2
s
3
0
−
⎣
⎦
≡
⎣
⎦
,
⎣
⎦
≡
⎣
⎦
,
Q
=
S
(
Q
)=
Ru
0
⎡
⎤
0
1
0
⎣
A
ρ
ψ
A
−
u
2
2
u
A
⎦
.
A
(
Q
)=
ρ
ψ
K
(2.13)
0
0
0
Next we study some mathematical properties of the equations.
