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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Consider the set of conservation equations :
l
i
(
A
i
j
∂
u
j
∂
t
+
a
i
j
∂
u
j
∂
x
)
+
l
j
b
j
=
0
{\displaystyle l_{i}\left(A_{ij}{\frac {\partial u_{j}}{\partial t}}+a_{ij}{\frac {\partial u_{j}}{\partial x}}\right)+l_{j}b_{j}=0}
where
A
i
j
{\displaystyle A_{ij}}
and
a
i
j
{\displaystyle a_{ij}}
are the elements of the matrices
A
{\displaystyle \mathbf {A} }
and
a
{\displaystyle \mathbf {a} }
where
l
i
{\displaystyle l_{i}}
and
b
i
{\displaystyle b_{i}}
are elements of vectors . It will be asked if it is possible to rewrite this equation to
m
j
(
β
∂
u
j
∂
t
+
α
∂
u
j
∂
x
)
+
l
j
b
j
=
0
{\displaystyle m_{j}\left(\beta {\frac {\partial u_{j}}{\partial t}}+\alpha {\frac {\partial u_{j}}{\partial x}}\right)+l_{j}b_{j}=0}
To do this curves will be introduced in the
(
x
,
t
)
{\displaystyle (x,t)}
plane defined by the vector field
(
α
,
β
)
{\displaystyle (\alpha ,\beta )}
. The term in the brackets will be rewritten in terms of a total derivative where
x
,
t
{\displaystyle x,t}
are parametrized as
x
=
X
(
η
)
,
t
=
T
(
η
)
{\displaystyle x=X(\eta ),t=T(\eta )}
d
u
j
d
η
=
T
′
∂
u
j
∂
t
+
X
′
∂
u
j
∂
x
{\displaystyle {\frac {du_{j}}{d\eta }}=T'{\frac {\partial u_{j}}{\partial t}}+X'{\frac {\partial u_{j}}{\partial x}}}
comparing the last two equations we find
α
=
X
′
(
η
)
,
β
=
T
′
(
η
)
{\displaystyle \alpha =X'(\eta ),\beta =T'(\eta )}
which can be now written in characteristic form
m
j
d
u
j
d
η
+
l
j
b
j
=
0
{\displaystyle m_{j}{\frac {du_{j}}{d\eta }}+l_{j}b_{j}=0}
where we must have the conditions
l
i
A
i
j
=
m
j
T
′
{\displaystyle l_{i}A_{ij}=m_{j}T'}
l
i
a
i
j
=
m
j
X
′
{\displaystyle l_{i}a_{ij}=m_{j}X'}
where
m
j
{\displaystyle m_{j}}
can be eliminated to give the necessary condition
l
i
(
A
i
j
X
′
−
a
i
j
T
′
)
=
0
{\displaystyle l_{i}(A_{ij}X'-a_{ij}T')=0}
so for a nontrivial solution is the determinant
|
A
i
j
X
′
−
a
i
j
T
′
|
=
0
{\displaystyle |A_{ij}X'-a_{ij}T'|=0}
For Riemann invariants we are concerned with the case when the matrix
A
{\displaystyle \mathbf {A} }
is an identity matrix to form
∂
u
i
∂
t
+
a
i
j
∂
u
j
∂
x
=
0
{\displaystyle {\frac {\partial u_{i}}{\partial t}}+a_{ij}{\frac {\partial u_{j}}{\partial x}}=0}
notice this is homogeneous due to the vector
n
{\displaystyle \mathbf {n} }
being zero. In characteristic form the system is
l
i
d
u
i
d
t
=
0
{\displaystyle l_{i}{\frac {du_{i}}{dt}}=0}
with
d
x
d
t
=
λ
{\displaystyle {\frac {dx}{dt}}=\lambda }
Where
l
{\displaystyle l}
is the left eigenvector of the matrix
A
{\displaystyle \mathbf {A} }
and
λ
′
s
{\displaystyle \lambda 's}
is the characteristic speeds of the eigenvalues of the matrix
A
{\displaystyle \mathbf {A} }
which satisfy
|
A
−
λ
δ
i
j
|
=
0
{\displaystyle |A-\lambda \delta _{ij}|=0}
To simplify these characteristic equations we can make the transformations such that
d
r
d
t
=
l
i
d
u
i
d
t
{\displaystyle {\frac {dr}{dt}}=l_{i}{\frac {du_{i}}{dt}}}
which form
μ
l
i
d
u
i
=
d
r
{\displaystyle \mu l_{i}du_{i}=dr}
An integrating factor
μ
{\displaystyle \mu }
can be multiplied in to help integrate this. So the system now has the characteristic form
d
r
d
t
=
0
{\displaystyle {\frac {dr}{dt}}=0}
on
d
x
d
t
=
λ
i
{\displaystyle {\frac {dx}{dt}}=\lambda _{i}}
which is equivalent to the diagonal system [2]
r
t
k
+
λ
k
r
x
k
=
0
,
{\displaystyle r_{t}^{k}+\lambda _{k}r_{x}^{k}=0,}
k
=
1
,
.
.
.
,
N
.
{\displaystyle k=1,...,N.}
The solution of this system can be given by the generalized hodograph method .[3] [4]
Example
[ edit ]
Consider the one-dimensional Euler equations written in terms of density
ρ
{\displaystyle \rho }
and velocity
u
{\displaystyle u}
are
ρ
t
+
ρ
u
x
+
u
ρ
x
=
0
{\displaystyle \rho _{t}+\rho u_{x}+u\rho _{x}=0}
u
t
+
u
u
x
+
(
c
2
/
ρ
)
ρ
x
=
0
{\displaystyle u_{t}+uu_{x}+(c^{2}/\rho )\rho _{x}=0}
with
c
{\displaystyle c}
being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form
(
ρ
u
)
t
+
(
u
ρ
c
2
ρ
u
)
(
ρ
u
)
x
=
(
0
0
)
{\displaystyle \left({\begin{matrix}\rho \\u\end{matrix}}\right)_{t}+\left({\begin{matrix}u&\rho \\{\frac {c^{2}}{\rho }}&u\end{matrix}}\right)\left({\begin{matrix}\rho \\u\end{matrix}}\right)_{x}=\left({\begin{matrix}0\\0\end{matrix}}\right)}
where the matrix
a
{\displaystyle \mathbf {a} }
from the analysis above the eigenvalues and eigenvectors need to be found. The eigenvalues are found to satisfy
λ
2
−
2
u
λ
+
u
2
−
c
2
=
0
{\displaystyle \lambda ^{2}-2u\lambda +u^{2}-c^{2}=0}
to give
λ
=
u
±
c
{\displaystyle \lambda =u\pm c}
and the eigenvectors are found to be
(
1
c
ρ
)
,
(
1
−
c
ρ
)
{\displaystyle \left({\begin{matrix}1\\{\frac {c}{\rho }}\end{matrix}}\right),\left({\begin{matrix}1\\-{\frac {c}{\rho }}\end{matrix}}\right)}
where the Riemann invariants are
r
1
=
J
+
=
u
+
∫
c
ρ
d
ρ
,
{\displaystyle r_{1}=J_{+}=u+\int {\frac {c}{\rho }}d\rho ,}
r
2
=
J
−
=
u
−
∫
c
ρ
d
ρ
,
{\displaystyle r_{2}=J_{-}=u-\int {\frac {c}{\rho }}d\rho ,}
(
J
+
{\displaystyle J_{+}}
and
J
−
{\displaystyle J_{-}}
are the widely used notations in gas dynamics ). For perfect gas with constant specific heats, there is the relation
c
2
=
const
γ
ρ
γ
−
1
{\displaystyle c^{2}={\text{const}}\,\gamma \rho ^{\gamma -1}}
, where
γ
{\displaystyle \gamma }
is the specific heat ratio , to give the Riemann invariants[5] [6]
J
+
=
u
+
2
γ
−
1
c
,
{\displaystyle J_{+}=u+{\frac {2}{\gamma -1}}c,}
J
−
=
u
−
2
γ
−
1
c
,
{\displaystyle J_{-}=u-{\frac {2}{\gamma -1}}c,}
to give the equations
∂
J
+
∂
t
+
(
u
+
c
)
∂
J
+
∂
x
=
0
{\displaystyle {\frac {\partial J_{+}}{\partial t}}+(u+c){\frac {\partial J_{+}}{\partial x}}=0}
∂
J
−
∂
t
+
(
u
−
c
)
∂
J
−
∂
x
=
0
{\displaystyle {\frac {\partial J_{-}}{\partial t}}+(u-c){\frac {\partial J_{-}}{\partial x}}=0}
In other words,
d
J
+
=
0
,
J
+
=
const
along
C
+
:
d
x
d
t
=
u
+
c
,
d
J
−
=
0
,
J
−
=
const
along
C
−
:
d
x
d
t
=
u
−
c
,
{\displaystyle {\begin{aligned}&dJ_{+}=0,\,J_{+}={\text{const}}\quad {\text{along}}\,\,C_{+}\,:\,{\frac {dx}{dt}}=u+c,\\&dJ_{-}=0,\,J_{-}={\text{const}}\quad {\text{along}}\,\,C_{-}\,:\,{\frac {dx}{dt}}=u-c,\end{aligned}}}
where
C
+
{\displaystyle C_{+}}
and
C
−
{\displaystyle C_{-}}
are the characteristic curves. This can be solved by the hodograph transformation . In the hodographic plane, if all the characteristics collapses into a single curve, then we obtain simple waves . If the matrix form of the system of pde's is in the form
A
∂
v
∂
t
+
B
∂
v
∂
x
=
0
{\displaystyle A{\frac {\partial v}{\partial t}}+B{\frac {\partial v}{\partial x}}=0}
Then it may be possible to multiply across by the inverse matrix
A
−
1
{\displaystyle A^{-1}}
so long as the matrix determinant of
A
{\displaystyle \mathbf {A} }
is not zero.
See also
[ edit ]
References
[ edit ]
^
Kamchatnov, A. M. (2000). Nonlinear Periodic Waves and their Modulations . World Scientific . ISBN 978-981-02-4407-1 .
^ Tsarev, S. P. (1985). "On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type" (PDF) . Soviet Mathematics - Doklady . 31 (3 ): 488–491. MR 2379468 . Zbl 0605.35075 . Archived from the original (PDF) on 2012-03-30. Retrieved 2011-08-20 .
^ Zelʹdovich, I. B., & Raĭzer, I. P. (1966). Physics of shock waves and high-temperature hydrodynamic phenomena (Vol. 1). Academic Press.
^ Courant, R., & Friedrichs, K. O. 1948 Supersonic flow and shock waves. New York: Interscience.
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