The eigensystem of the Euler equations
PDF of note
In this document I list the eigensystem of the Euler equations valid
for a general equation of state. The formulas are taken from
[Kulikovskii2001], Chapter 3, section 3.1. The Euler equations can be
written in conservative form as
(1)\[\begin{split}\frac{\partial}{\partial{t}}
\left[
\begin{matrix}
\rho \\
\rho u \\
\rho v \\
\rho w \\
E
\end{matrix}
\right]
+
\frac{\partial}{\partial{x}}
\left[
\begin{matrix}
\rho u \\
\rho u^2 + p \\
\rho uv \\
\rho uw \\
(E+p)u
\end{matrix}
\right]
=
0\end{split}\]
where
\[E = \rho \varepsilon + \frac{1}{2}\rho q^2\]
is the total energy and \(\varepsilon\) is the internal energy of
the fluid and \(q^2=u^2 + v^2 + w^2\). The pressure is given by an
equation of state (EOS) \(p=p(\varepsilon, \rho)\). For an ideal
gas the EOS is \(p = (\gamma-1)\rho \varepsilon\).
The eigenvalues are \(\{u-c, u, u, u, u+c\}\). The right
eigenvectors of the flux Jacobian are given by the columns of the
matrix
(2)\[\begin{split}R
=
\left[
\begin{matrix}
1 & 0 & 0 & 1 & 1 \\
u-c & 0 & 0 & u & u+c \\
v & 1 & 0 & v & v \\
w & 0 & 1 & w & w \\
h-uc & v & w & h-c^2/b & h+uc
\end{matrix}
\right]
\label{eq:rev}\end{split}\]
here
\[\begin{split}h &= (E+p)/\rho \\
c &= \sqrt{\frac{\partial p}{\partial \rho}
+ \frac{p}{\rho^2}\frac{\partial p}{\partial \varepsilon}}\end{split}\]
is the enthalpy and the sound speed respectively. Also,
\[\begin{split}b &= \frac{1}{\rho}\frac{\partial p}{\partial \varepsilon}.\end{split}\]
Note that for ideal gas EOS we have
\[\begin{split}h &= \frac{c^2}{\gamma-1} + \frac{1}{2}q^2 \\
c &= \sqrt{\frac{\gamma p}{\rho}}\end{split}\]
and \(b=\gamma-1\). Hence, in this case the term \(h-c^2/b\)
in (2) is just \(q^2/2\). The left eigenvectors are the
rows of the matrix
(3)\[\begin{split}L
=
\frac{b}{2c^2}
\left[
\begin{matrix}
\theta+uc/b & -u-c/b & -v & -w & 1 \\
-2vc^2/b & 0 & 2c^2/b & 0 & 0 \\
-2wc^2/b & 0 & 0 & 2c^2/b & 0 \\
2h-2q^2 & 2u & 2v & 2w & -2 \\
\theta-uc/b & -u+c/b & -v & -w & 1
\end{matrix}
\right]\end{split}\]
where
\[\theta = q^2 - \frac{E}{\rho}
+ \rho\frac{\partial p / \partial \rho}{\partial p / \partial \varepsilon}\]
which, for an ideal gas EOS reduces to \(q^2/2\).
Now consider the problem of splitting a jump vector \(\Delta
\equiv [\delta_0,\delta_1,\delta_2,\delta_3,\delta_4]^T\) into
coefficients neeeded in computing the Riemann problem. The
coefficients are given by \(L\Delta\). For an ideal gas law EOS,
after some algebra we can show
that an efficient way to compute these are
(4)\[\begin{split}\alpha_3 &= \frac{\gamma-1}{c^2}
\left[
(h-q^2)\delta_0 + u\delta_1 + v\delta_2 + w\delta_3 -\delta_4
\right] \\
\alpha_1 &= -v\delta_0 + \delta_2 \\
\alpha_2 &= -w\delta_0 + \delta_3 \\
\alpha_4 &= \frac{1}{2c}
\left[
\delta_1 + (c-u)\delta_0 - c\alpha_3
\right] \\
\alpha_0 &= \delta_0 - \alpha_3 - \alpha_4.\end{split}\]
References
| [Kulikovskii2001] | Andrei G. Kulikoviskii and Nikolai V. Pogorelov
and Andrei Yu. Semenov, Mathematical Aspects of Numerical
Solutions of Hyperbolic Systems, Chapman and Hall/CRC, 2001. |