# Euler Equation

## Conservation Form
The one dimensional version of the Euler equation for gas dynamics:
\begin{equation}
  \mathbf{u}_t + \mathbf{f}(\mathbf{u})_x = 0,\,\, t \geq 0, x \in \mathbb{R},
\end{equation}
where
\begin{equation}
  \mathbf{u} = \left(\begin{array}{c}\rho\\m\\E\end{array}\right),\,\, \mathbf{f}(\mathbf{u}) = \left(\begin{array}{c}m\\\rho v^2 + p\\(E+p)v\end{array}\right),
\end{equation}
with
\begin{equation*}
  m = \rho v, \,\, E = \frac{1}{2}\rho v^2 + \rho e,
\end{equation*}
where $\rho$ is the density, $v$ is the velocity, $m$ is the momentum, $E$ is the total energy, $e$ is the internal, and the pressure $p$ can be obtained from the equation of state.  For instance, for the perfect gas,
\begin{equation*}
  p(\mathbf{u}) = (\gamma -1)\rho e = (\gamma-1)\left(E - \frac{1}{2}\frac{m^2}{\rho}\right)
\end{equation*}
It is easy to calculate that
\begin{equation*}
    \frac{\partial p}{\partial \rho} = \frac{1}{2}(\gamma-1)v^2,\,\, \frac{\partial p}{\partial m} = -(\gamma-1)v,\,\, \frac{\partial p}{\partial E} = \gamma - 1,
\end{equation*}
and the Jacobian of $\mathbf{f}(\mathbf{u})$ is given by
\begin{equation*}
    \mathbf{f}'(\mathbf{u}) = \left(\begin{array}{ccc}0 & 1 & 0 \\ \frac{1}{2}(\gamma-3)v^2 & (3-\gamma)v & \gamma-1 \\ \frac{1}{2}(\gamma-1)v^3-v(E+p)/\rho & (E+p)/\rho - (\gamma-1)v^2 & \gamma v\end{array}\right)
\end{equation*}
To get the eignvalue of $\mathbf{f}'(\mathbf{u})$, we have 
\begin{equation*}
    \det(\lambda I - \mathbf{f}') = \lambda^3 -3v\lambda^2 + (3v^2-c^2)\lambda + v(v^2 - c^2)= (\lambda - v)(\lambda - v-c)(\lambda-v+c),
\end{equation*}
where $c = \sqrt{\frac{\gamma p}{\rho}}$.


```python

```

8a214cb083fdde178bf2e01ca663941b135dc364
ipynb
Jupyter Notebook
Numerical_Methods/EulerEquation.ipynb
xiaozhouli/Jupyter
68d5a384dd939b3e8079da4470d6401d11b63a4c
MIT
2020-02-27T13:09:06.000Z
2021-11-14T09:50:30.000Z
Numerical_Methods/EulerEquation.ipynb
xiaozhouli/Jupyter
68d5a384dd939b3e8079da4470d6401d11b63a4c
MIT
Numerical_Methods/EulerEquation.ipynb
xiaozhouli/Jupyter
68d5a384dd939b3e8079da4470d6401d11b63a4c
MIT
2018-10-18T10:20:56.000Z
2021-09-24T08:09:27.000Z
Qwen/Qwen-72B
1. YES
2. YES