# The eigensystem of the Euler equations¶

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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.