Analytical solution of the Euler equations in 1D?

[Morridini]

Hi.

Since these equations are approaching three hundred years old I'm pretty sure someone must have solved them somewhere before. However I have not been able to find any text-books or papers that actually show me how to solve these equations. So I'm wondering if anyone here know where I can find an approach to the analytical solutions, or who simply know how I should attack a problem set like this?

Here are the equations:

\begin{align}
\frac{\partial\rho}{\partial t} + \frac{\partial}{\partial x}(\rho v) &= 0 \\
\frac{\partial}{\partial t}(\rho v) + \frac{\partial}{\partial x}(\rho v^2) +\frac{\partial p}{\partial x} &= - \rho\frac{\partial \phi}{\partial x} \\
\frac{\partial^2\phi}{\partial x^2}& = 4\pi G\rho
\end{align}

Were $t$ and $x$ are the space and time coordinates, $\rho = \rho(x,t)$ is a density, $v=v(x,t)$ is a velocity, $p=p(\rho)$ is the pressure related to the density with a constant $p=w\rho$ and $\phi=\phi(x,t)$ is a gravitational potential. The physical variables isn't really important here, as I am more curious as to the mathematical approach to find one analytical solution.

A general analytical solution probably doesn't exist for this set, but I believe I can give some initial conditions:
\begin{align}
\rho_0 = 1,\quad v_0 = sin(2\pi x), \quad p_0= 10^{-5}
\end{align}

Anyone got any sources or tips for solving this? Having multiple variables ($\rho$, $v$) as well as several partial derviatives ($\partial x, \partial t$) in the same equations is what is putting me off.

Thanks in advance for any help.

P.S.
Wasn't sure of where to put this thread, but decided for Math as the physics is kinda irrelevant at this point I think.

Cheers

EDIT: Oh, I see writing TeX in the text is not how it is in LaTeX on this forum, how should I do it?

 

[the_wolfman]

Euler's equation and the Navier-Stokes equations are incredibly difficult to solve, even in 1-D.

A simplified toy model that is often studied is the inviscid Burger's equation.

$d_t u + u d_x u = 0$

You can construct solutions to the Burger's equation using the method of characteristics. I know its a far cry from Euler's equation, but its a good place to start. However, don't be surprised if you have to resort to a numerical solution.

 

[Morridini]

Thanks for the suggestions, in the time after I had first written the original post and you answered I had come across the Method of Charecteristics, and I tried it on the equations posted above. Let me type them up here, as I am not quite sure of how to proceed.

EDIT: I seem to have forgotten pressure in what follows below... oh well, let's ignore pressure for now.

In the method of charatceristics I try to build two equations on the form:

\begin{align}
\frac{d\rho}{ds} &= \frac{\partial\rho}{\partial t}\frac{\partial t}{\partial s} + \frac{\partial\rho}{\partial x}\frac{\partial x}{\partial s} \\
\frac{dv}{ds} &= \frac{\partial v}{\partial t}\frac{\partial t}{\partial s} + \frac{\partial v}{\partial x}\frac{\partial x}{\partial s}
\end{align}

I then proceed to compare these to the Euler equations above and see that:

\begin{align}
\frac{\partial t}{\partial s} &= 1, \;\qquad\qquad \frac{\partial x}{\partial s} = v \\
\frac{d\rho}{ds} &= -\rho\frac{\partial v}{\partial x}, \qquad \frac{d v}{ds} = -\frac{\partial \phi}{\partial x}
\end{align}

Now at this point I am starting to struggle (keep in mind I just discovered this method (or actually, I vaguely remember doing something like this, just with matrices in a course once)), first of all I need initial conditions for $x(s=0)$ and $t(s=0)$.

On the other hand, I now seem to have two ordinary differential equations, one for ρ and v... Idle thoguhts suggest taking the spatial derivative of the last equation to get rid of the gravitational potential and introducing a ρ, and then taking the derivative of the former equation with regard to $s$ and combining them to get:

\begin{align}
\frac{\partial^2 \rho}{\partial s^2} - \frac{1}{\rho}\left(\frac{\partial \rho}{\partial s}\right)^2 - \rho^2 = 0
\end{align}

However before I continue along these lines I've to check to see if I haven't made any errors so far, do you agree with what I have done wolfman?

Also, I don't want to simplify the equations down any further as these are already simplifications of what I intend to solve (I'm on the hunt for these solutions in an expanding universe, so I'll have some terms of the scale factor and Hubble parameter cropping up as well.

Also 2, numerical solution isn't the way to go. That's because this is supposed to give me the analytical solution to compare numerical solutions I have already calculated to...

 

[the_wolfman]

It's been a while since I used the method of characteristics.

Again I want to stress that Eulers equation is incredibly hard to solve analytically. I suspect that you'll need to further simplify your equation if you want to compare it to analytic solutions.

Two cases that I would consider are the case of large gravity, and the case of weak gravity. (I'm ignoring gravity).

Start by defining the ratio $\frac{G\rho L^2}{V^2}$ where L is a characteristic length scale of your system.
In the limit where this ratio is large then the gravity term dominates the advective term and your equation becomes
$d_t u = -d_x \phi$

The other limit where the ratio is small allows you to ignore the gravity term. Then your equation becomes
$d_t u + u d_x u =0$

 

[Morridini]

I see what you're saying, but I have to give this some careful thought. The purpose of doing this is to study the effects of certain modified gravity theories on the baryon density in galaxy clusters... so I can't really ignore gravity. Sadly I'm in a bi of a rush now, but I'll give it some more thought. Thanks for all the input.