Smooth solution to the isentropic Euler equations

Question

In order to check the order of accuracy of a numerical method, I'd like to find a smooth (at least $C^2$) solution to the isentropic Euler system. This system reads as follows:

$$ \left\{ \begin{array}{lcl} \partial_t \rho + \partial_x ( \rho u ) &=& 0, \\ \partial_t ( \rho u ) + \partial_x ( \rho u^2 ) + \partial_x p(\rho) &=& 0, \end{array} \right. $$

where $\rho > 0$ is the density, $u$ is the velocity and $p(\rho) = \rho^\gamma$ the pressure, with $\gamma \in [1,3]$ (although I do not require the exact solution to be valid for this whole range, a single value of $\gamma$ would be enough).

I've been playing around with the equations, but to no avail. I also haven't managed to find a paper or book where this would have been done.

Thanks!

Answer 1

I found an answer in this article, Section 3.3.

It consists in noting that the Riemann invariants $ J_\pm = u \mp \frac 2 {\gamma - 1} c $, where $c = \sqrt \gamma \rho^{(\gamma - 1)/2}$ is the sound speed, are constant on the characteristic curves

$ \left\{ \begin{array}{l} dy / dt (x,t) = u \pm c, \\ y(x,0) = x. \end{array} \right. $

Using the expressions of the Riemann invariants and taking $\gamma = 3$, the characteristic curves read:

$ \left\{ \begin{array}{l} dy / dt (x,t) = J_\mp, \\ y(x,0) = x, \end{array} \right. $

which means that both Riemann invariants satisfy a Burgers equation, as follows:

$ \left\{ \begin{array}{l} \partial_t J_+ + J_+ \partial J_+ = 0, \\ \partial_t J_- + J_- \partial J_- = 0. \end{array} \right. $

Solving these equations yields $J_\pm$, and thus yields $\rho$ and $u$. A nonlinear equation solver, such as Newton's method, will be required. For small enough time, the solution is as smooth as the initial data.