4. 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,

\[b = \frac{1}{\rho}\frac{\partial p}{\partial \varepsilon}.\]

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)\[\begin{split}\alpha_3 &= \frac{\gamma-1}{c^2} \left[ (h-q^2)\delta_0 + u\delta_1 + v\delta_2 + w\delta_3 -\delta_4 \right] \\ \alpha_1 &= -v\delta_0 + \delta_2 \\ \alpha_2 &= -w\delta_0 + \delta_3 \\ \alpha_4 &= \frac{1}{2c} \left[ \delta_1 + (c-u)\delta_0 - c\alpha_3 \right] \\ \alpha_0 &= \delta_0 - \alpha_3 - \alpha_4.\end{split}\]

4.1. 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.