# Odd result from an eigenvalue problem in the Euler equations

1. Aug 3, 2010

### thezealite

Given the Euler equations in two dimensions in a moving reference frame:

$$\frac{\partial U}{\partial t} + \frac{\partial F\left(U\right)}{\partial x} = 0$$

$$U = \left(\rho , \rho u , \rho v , \rho e \right)$$

$$F\left(U\right) = \left(\left(1-h\right)\rho u , \left(1-h\right)\rho u^2 + p , \left(1-h\right)\rho u v , \left(1-h\right)\rho u e + u p \right)$$

$$p = \left(\gamma -1\right)\rho \left(e-\frac{1}{2}\left(u^2+v^2\right)\right)$$

Where h accounts for relative motion, the eigenvalues and eigenvectors of the system are reported to be

$$\Lambda = \left(\left(1-h\right) u,\left(1-h\right) u, \left(1-h\right)u+a,\left(1-h\right)u-a\right)$$

and

$$EV1=\left(0,1,0,0\right)\ ,\ EV2=\left(0,0,0,1\right)$$

$$EV3=\left(\frac{1}{a^2},1,\frac{1}{\rho a} ,0\right)$$

$$EV4=\left(\frac{1}{a^2},1,-\frac{1}{\rho a} ,0\right)$$

where

$$a=\sqrt{\gamma\frac{p}{\rho}$$

I have tried to reproduce this, and I've also tried to reverse it, but I'm not having any luck. Is this reported result just wrong? or am I missing something simple?