Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant. They were first obtained by Bernhard Riemann in his work on plane waves in gas dynamics.[1]
Consider the set of conservation equations:
li\left(Aij
| \partialuj | |
| \partialt |
+aij
| \partialuj | |
| \partialx |
\right)+ljbj=0
where
Aij
aij
A
a
li
bi
| m | +\alpha | ||||
|
| \partialuj | |
| \partialx |
\right)+ljbj=0
To do this curves will be introduced in the
(x,t)
(\alpha,\beta)
x,t
x=X(η),t=T(η)
| duj | =T' | |
| dη |
| \partialuj | +X' | |
| \partialt |
| \partialuj | |
| \partialx |
comparing the last two equations we find
\alpha=X'(η),\beta=T'(η)
which can be now written in characteristic form
| m | ||||
|
+ljbj=0
where we must have the conditions
liAij=mjT'
liaij=mjX'
where
mj
li(AijX'-aijT')=0
so for a nontrivial solution is the determinant
|AijX'-aijT'|=0
For Riemann invariants we are concerned with the case when the matrix
A
| \partialui | |
| \partialt |
+aij
| \partialuj | |
| \partialx |
=0
notice this is homogeneous due to the vector
n
| l | ||||
|
=0
| dx | |
| dt |
=λ
Where
l
A
λ's
A
|A-λ\deltaij|=0
To simplify these characteristic equations we can make the transformations such that
| dr | |
| dt |
| =l | ||||
|
which form
\mulidui=dr
\mu
| dr | |
| dt |
=0
| dx | |
| dt |
=λi
which is equivalent to the diagonal system[2]
| k | |
| r | |
| t |
+λkr
| k=0, | |
| x |
k=1,...,N.
The solution of this system can be given by the generalized hodograph method.[3] [4]
Consider the one-dimensional Euler equations written in terms of density
\rho
u
\rhot+\rhoux+u\rhox=0
ut+uu
| 2/\rho)\rho | |
| x=0 |
with
c
\left(\begin{matrix}\rho\ u\end{matrix}\right)t+\left(\begin{matrix}u&\rho\
| c2 | |
| \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
λ2-2uλ+u2-c2=0
to give
λ=u\pmc
and the eigenvectors are found to be
\left(\begin{matrix}1\
| c | |
| \rho |
\end{matrix}\right),\left(\begin{matrix}1\ -
| c | |
| \rho |
\end{matrix}\right)
where the Riemann invariants are
r1=J+=u+\int
| c | |
| \rho |
d\rho,
r2=J-=u-\int
| c | |
| \rho |
d\rho,
(
J+
J-
c2=const\gamma\rho\gamma-1
\gamma
| J | ||||
|
c,
| J | ||||
|
c,
to give the equations
| \partialJ+ | +(u+c) | |
| \partialt |
| \partialJ+ | |
| \partialx |
=0
| \partialJ- | +(u-c) | |
| \partialt |
| \partialJ- | |
| \partialx |
=0
In other words,
\begin{align} &dJ+=0,J+=const alongC+:
| dx | |
| dt |
=u+c,\\ &dJ-=0,J-=const alongC-:
| dx | |
| dt |
=u-c, \end{align}
where
C+
C-
| A | \partialv | +B |
| \partialt |
| \partialv | |
| \partialx |
=0
Then it may be possible to multiply across by the inverse matrix
A-1
A