Biomedical Engineering Reference
In-Depth Information
2.4.4 Solution of Riemann problem
In previous sections we have put in place all the necessary relations to obtain the
solution of the Riemann problem in the Star Region, which is the region in physical
space located between the waves associated with the outer characteristic fields. The
procedure to find the solution is embodied in the following proposition.
Proposition 4.6 (solution of Riemann problem).
The solution of the Riemann prob-
lem in the
Star Region
is given by the solution of the following non-linear system
⎫
⎬
f
1
(
x
1
,
x
2
)
=
x
2
−
u
L
+
f
L
(
x
1
)
=
0
,
f
2
(
x
1
,
x
2
,
x
3
,
x
4
)=
x
2
x
1
−
x
4
x
3
=
0
,
K
L
x
1
A
0
L
m
1
K
R
x
3
A
0
R
m
1
1
x
2
−
x
4
)+
f
3
(
x
1
,
x
2
,
x
3
,
x
4
)=
2
ρ
(
−
−
−
=
0
,
(2.47)
f
4
(
x
3
,
x
4
)
=
x
4
−
u
R
−
f
R
(
x
3
)
=
0
,
where the unknowns of the problem are
X
=[
x
1
,
x
2
,
x
3
,
x
4
]
≡
[
A
∗
L
,
u
∗
L
,
A
∗
R
,
u
∗
R
]
,
(2.48)
with
⎧
⎨
B
L
(
x
1
−
A
L
)(
x
m
+
1
1
A
m
+
1
L
−
)
if
A
∗
L
>
A
L
,
A
L
x
1
f
L
(
x
1
)=
(2.49)
m
D
L
x
m
/
2
c
L
⎩
2
1
−
if
A
∗
L
≤
A
L
,
and
⎧
⎨
B
R
(
x
3
−
A
R
)(
x
m
+
1
3
A
m
+
1
R
−
)
if
A
∗
R
>
A
R
,
A
R
x
3
f
R
(
x
3
)=
(2.50)
m
D
R
x
m
/
2
c
R
⎩
2
−
if
A
∗
R
≤
A
R
,
3
mK
L
ρ
mK
R
ρ
D
L
=
A
0
L
,
D
R
=
A
0
R
.
(2.51)
The wave speeds
c
L
and
c
R
are evaluated on the data according to (2.15). The con-
stants
K
L
and
K
R
are evaluated on the data from to (2.6) and
B
is given by (2.28).
Proof.
The proof involves putting together the results stated previously. Details are
omitted.
Remarks:
•
Complete solution.
The numerical solution of the non-linear system (2.47) gives
the four unknowns in the
Star Region
. The rest of the solution follows by applying
the wave relations studied in the previous sections. Part of the process of finding
the complete solution involves a procedure for the solution at a point inside a
rarefaction fan. Details of this are given in Sect. 6.
