Riemann invariant

Consider the set of conservation equations:

${\displaystyle l_{i}\left(A_{ij}{\frac {\partial u_{j}}{\partial t}}+a_{ij}{\frac {\partial u_{j}}{\partial x}}\right)+l_{j}b_{j}=0}$

where ${\displaystyle A_{ij}}$ and ${\displaystyle a_{ij}}$ are the elements of the matrices ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {a} }$ where ${\displaystyle l_{i}}$ and ${\displaystyle b_{i}}$ are elements of vectors. It will be asked if it is possible to rewrite this equation to

${\displaystyle m_{j}\left(\beta {\frac {\partial u_{j}}{\partial t}}+\alpha {\frac {\partial u_{j}}{\partial x}}\right)+l_{j}b_{j}=0}$

To do this curves will be introduced in the ${\displaystyle (x,t)}$ plane defined by the vector field ${\displaystyle (\alpha ,\beta )}$. The term in the brackets will be rewritten in terms of a total derivative where ${\displaystyle x,t}$ are parametrized as ${\displaystyle x=X(\eta ),t=T(\eta )}$

${\displaystyle {\frac {du_{j}}{d\eta }}=T'{\frac {\partial u_{j}}{\partial t}}+X'{\frac {\partial u_{j}}{\partial x}}}$

comparing the last two equations we find

${\displaystyle \alpha =X'(\eta ),\beta =T'(\eta )}$

which can be now written in characteristic form

${\displaystyle m_{j}{\frac {du_{j}}{d\eta }}+l_{j}b_{j}=0}$

where we must have the conditions

${\displaystyle l_{i}A_{ij}=m_{j}T'}$

${\displaystyle l_{i}a_{ij}=m_{j}X'}$

where ${\displaystyle m_{j}}$ can be eliminated to give the necessary condition

${\displaystyle l_{i}(A_{ij}X'-a_{ij}T')=0}$

so for a nontrivial solution is the determinant

${\displaystyle |A_{ij}X'-a_{ij}T'|=0}$

For Riemann invariants we are concerned with the case when the matrix ${\displaystyle \mathbf {A} }$ is an identity matrix to form

${\displaystyle {\frac {\partial u_{i}}{\partial t}}+a_{ij}{\frac {\partial u_{j}}{\partial x}}=0}$

notice this is homogeneous due to the vector ${\displaystyle \mathbf {n} }$ being zero. In characteristic form the system is

${\displaystyle l_{i}{\frac {du_{i}}{dt}}=0}$ with ${\displaystyle {\frac {dx}{dt}}=\lambda }$

Where ${\displaystyle l}$ is the left eigenvector of the matrix ${\displaystyle \mathbf {A} }$ and ${\displaystyle \lambda 's}$ is the characteristic speeds of the eigenvalues of the matrix ${\displaystyle \mathbf {A} }$ which satisfy

${\displaystyle |A-\lambda \delta _{ij}|=0}$

To simplify these characteristic equations we can make the transformations such that ${\displaystyle {\frac {dr}{dt}}=l_{i}{\frac {du_{i}}{dt}}}$

which form

${\displaystyle \mu l_{i}du_{i}=dr}$

An integrating factor ${\displaystyle \mu }$ can be multiplied in to help integrate this. So the system now has the characteristic form

${\displaystyle {\frac {dr}{dt}}=0}$ on ${\displaystyle {\frac {dx}{dt}}=\lambda _{i}}$

which is equivalent to the diagonal system[2]

${\displaystyle r_{t}^{k}+\lambda _{k}r_{x}^{k}=0,}$ ${\displaystyle k=1,...,N.}$

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 ${\displaystyle \rho }$ and velocity ${\displaystyle u}$ are

${\displaystyle \rho _{t}+\rho u_{x}+u\rho _{x}=0}$

${\displaystyle u_{t}+uu_{x}+(c^{2}/\rho )\rho _{x}=0}$

with ${\displaystyle c}$ being the speed of sound is introduced on account of isentropic assumption. Write this system in matrix form

${\displaystyle \left({\begin{matrix}\rho \\u\end{matrix}}\right)_{t}+\left({\begin{matrix}u&\rho \\{\frac {c^{2}}{\rho }}&u\end{matrix}}\right)\left({\begin{matrix}\rho \\u\end{matrix}}\right)_{x}=\left({\begin{matrix}0\\0\end{matrix}}\right)}$

where the matrix ${\displaystyle \mathbf {a} }$ from the analysis above the eigenvalues and eigenvectors need to be found. The eigenvalues are found to satisfy

${\displaystyle \lambda ^{2}-2u\lambda +u^{2}-c^{2}=0}$

to give

${\displaystyle \lambda =u\pm c}$

and the eigenvectors are found to be

${\displaystyle \left({\begin{matrix}1\\{\frac {c}{\rho }}\end{matrix}}\right),\left({\begin{matrix}1\\-{\frac {c}{\rho }}\end{matrix}}\right)}$

where the Riemann invariants are

${\displaystyle r_{1}=J_{+}=u+\int {\frac {c}{\rho }}d\rho ,}$

${\displaystyle r_{2}=J_{-}=u-\int {\frac {c}{\rho }}d\rho ,}$

(${\displaystyle J_{+}}$ and ${\displaystyle J_{-}}$ are the widely used notations in gas dynamics). For perfect gas with constant specific heats, there is the relation ${\displaystyle c^{2}={\text{const}}\,\gamma \rho ^{\gamma -1}}$, where ${\displaystyle \gamma }$ is the specific heat ratio, to give the Riemann invariants[5][6]

${\displaystyle J_{+}=u+{\frac {2}{\gamma -1}}c,}$

${\displaystyle J_{-}=u-{\frac {2}{\gamma -1}}c,}$

to give the equations

${\displaystyle {\frac {\partial J_{+}}{\partial t}}+(u+c){\frac {\partial J_{+}}{\partial x}}=0}$

${\displaystyle {\frac {\partial J_{-}}{\partial t}}+(u-c){\frac {\partial J_{-}}{\partial x}}=0}$

In other words,

${\displaystyle {\begin{aligned}&dJ_{+}=0,\,J_{+}={\text{const}}\quad {\text{along}}\,\,C_{+}\,:\,{\frac {dx}{dt}}=u+c,\\&dJ_{-}=0,\,J_{-}={\text{const}}\quad {\text{along}}\,\,C_{-}\,:\,{\frac {dx}{dt}}=u-c,\end{aligned}}}$

where ${\displaystyle C_{+}}$ and ${\displaystyle C_{-}}$ 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

${\displaystyle A{\frac {\partial v}{\partial t}}+B{\frac {\partial v}{\partial x}}=0}$

Then it may be possible to multiply across by the inverse matrix ${\displaystyle A^{-1}}$ so long as the matrix determinant of ${\displaystyle \mathbf {A} }$ is not zero.

• Simple wave