Biomedical Engineering Reference
In-Depth Information
Integration in phase space gives
c
(
A
)
dA
+
u
=
constant
.
A
After expressing
c
explicitly as a function of
A
and performing exact integration
leads to the second result in (2.21).
(
A
)
Proposition 3.6.
The generalized Riemann invariants for
λ
3
=
u
+
c
are
2
m
c
=
,
−
u
=
constant
.
(2.23)
K
constant
Proof.
The proof is entirely analogous to the previous case and is thus omitted.
2.4 The Riemann problem
Here we pose and solve exactly the Riemann problem for system (2.12), that is the
special Cauchy problem with piece-wise constant initial condition, namely
⎫
⎬
+
(
)
∂
=
,
∈R ,
>
,
∂
t
Q
A
Q
x
Q
0
x
t
0
⎧
⎨
Q
L
if
x
<
0
,
(2.24)
⎭
(
,
)=
Q
x
0
⎩
Q
R
if
x
>
0
.
The structure of the similarity solution of the problem is shown in Fig. 2.2 in the
entire
x
-
t
half plane. There are three wave families. The left family is associated with
the eigenvalue
λ
1
, the middle family is associated with
λ
2
and the right wave family
is associated with
λ
3
. Waves associated with the
genuinely non-linear
characteristic
fields
3
are either shocks (discontinuous solutions) or rarefactions (smooth
solutions), while the wave associated with the
linearly degenerate
characteristic field
λ
λ
1
and
λ
2
is a contact discontinuity. The entire solution consists of four constant states,
namely
Q
L
(data),
Q
∗
L
,
Q
∗
R
and
Q
R
(data), separated by three waves. The unknown
states to be found are
Q
∗
L
(left of the contact) and
Q
∗
R
(right of the contact). If any
of the
λ
3
waves is a rarefaction then there will be a smooth transition between
two adjacent constant states. In order to solve exactly the entire initial-value problem
we need to establish appropriate jump conditions across each characteristic field to
connect the unknown states
Q
∗
L
and
Q
∗
R
to the initial conditions
Q
L
(left) and
Q
R
(right) respectively. In what follows we establish such jump conditions across each
characteristic field.
λ
1
and
