What I'm looking for:

Let $\vec{W}$ be the vector of conserved variables for a 1-dimensional, adiabatic, (special) relativistic, electrically neutral fluid. (Yes, something that simple!) I'm looking for a paper that derives the form of the matrix $A$ that linearizes the evolution equation. That is, $$ \partial_t \vec{W} \approx A \partial_x \vec{W}. $$ Alternatively, the matrix $B$ that does the same for the conserved variables will work: $$ \partial_t \vec{U} \approx B \partial_x \vec{U}. $$

These matrices are useful in fluid computations for several reasons. I need more than just the eigenvalues (used in certain solvers) - I also need the eigenfunctions. I've found plenty of papers that treat the nonrelativistic case (for one of many, many examples, see the appendices of Stone et al. 2008, ApJS 178 137) both with and without magnetism. I've found some papers that just quote a few eigenfunctions for relativistic MHD, but these are often the ones that are interesting only with magnetism in play.

I'm looking for whatever paper derives these matrices in the rather simple case I'm dealing with. An answer that gives the derivation would be nice, but I'm also trying to locate the relevant literature. In particular, I would like a paper addresses the physical reliability/usefulness of the linear approximation.


There are three primitive variables defining my fluid: rest-mass density $\rho$, velocity $v$, and pressure $p$. By convention, these variables are combined into a vector $\vec{W} = (\rho, v, p)^\mathrm{T}$.

Many approaches to evolving fluids deal with the equations in flux-conservative form. Here, the conserved variables are \begin{align} D & = \gamma\rho && \text{(lab-frame density),} \\ M & = Dh\gamma v && \text{(relativistic momentum),} \\ E & = Dh\gamma - p && \text{(relativistic energy).} \end{align} Here I define \begin{align} \gamma & = \frac{1}{\sqrt{1-v^2}} && \text{(standard Lorentz factor),} \\ h & = 1 + \frac{\Gamma}{\Gamma-1} \left(\frac{p}{\rho}\right) && \text{(enthalpy),} \end{align} where the ratio of specific heats $\Gamma$ is assumed to be constant. These are often combined as $\vec{U} = (D, M, E)^\mathrm{T}$.

Along with the vector of conserved quantities, we can define the vector of fluxes $$ \vec{F} = \begin{pmatrix} Dv \\ Mv + p \\ M \end{pmatrix}. $$ Then we have the relation $$ \partial_t \vec{U} + \partial_x \vec{F} = 0, $$ which is used as the basis for most Riemann solvers and many fluid codes in general.

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    $\begingroup$ I think this 2009 paper by Aton et al might help. $\endgroup$ – Kyle Kanos Nov 5 '13 at 23:14
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    $\begingroup$ Could this paper be of help? Chen, J. (1997). Conservation laws for relativistic fluid dynamics. Archive for Rational Mechanics and Analysis, 139(4), 377-398. online version $\endgroup$ – user23660 Nov 6 '13 at 2:20
  • $\begingroup$ @KyleKanos Thanks! That's a nice thorough overview of relativistic MHD. I imagine all I can just set some values to 0 to recover the non-magnetic result, and I get the more advanced stuff along for the ride. $\endgroup$ – user10851 Nov 6 '13 at 3:14
  • $\begingroup$ @user23660 Indeed that's just what I'm looking for. It seems the author switches from $(\rho, v, p)$ space to $(\rho, v, s)$ (which makes sense, given that one of the eigenvectors is an entropy wave), but that should be easy enough to undo. If either you or Kyle want to post as a full answer, feel free to do so. $\endgroup$ – user10851 Nov 6 '13 at 3:17

Essentially, you want to solve the Riemann problem for the conservative system of RHD. For a general equation of state this is pretty tricky. For a general equation of state there are two real methods of doing this, both are described in An upwind numerical scheme for relatavistic hydrodynamics with a general equation of state.

This is one of many papers which I studied in great depth when creating some of my own schemes for General Relativistic Magnetohydrodynamics. S. S. Komissarov is a leading guy in this area (and was my Ph.D. supervisor) so I would check out some of his other works on numerical schemes for relativistic fluids.

If you have any further questions about the the methods used or the derivations of the eigenvectors/values, feel free to ask.

I hope this helps.

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