# Euler Equations, Sod shock tube & conservation

1. Jun 27, 2014

### Antigravity324

Is momemtum conserved?
I am considering the Euler equations in conservative form and solving the Sod shock tube problem I have written a Godunov finite volume type solver. It solves for density ρ, momentum ρu, and total energy E; therefore, I would expect all of these quantities to be conserved wrt time. Density and total energy are indeed conserved, however, momentum is not.
The Euler equations are given by
$$\begin{bmatrix} \rho \\ \rho u \\ E \end{bmatrix}_t + \begin{bmatrix} \rho u \\ \rho u^2 + p \\ u(E +p) \end{bmatrix}_x = 0$$
where pressure is related to the conserved quantities by
$$p = (\gamma - 1)(E - 0.5 \rho u^2)$$
The Sod shock problem splits the domain into two regions separated by a density and pressure discontinuity with initial velocity zero. That is, $$\rho_L = 1, \rho_R = .125; p_L=1, p_R =.1; u_L=u_R =0$$ These initial conditions imply that ρu=0, therefore, momentum should be zero throughout the simulation.

The solution profiles are well known and can be found here or here. We see that ρ >0 and u > 0, therefore, there is no way for momentum ρu = 0 (which it should be from the intial conditions). As a result, I do not even see why it is reasonable to expect that momentum would be conserved. A paper by Sod himself which surveys some methods for solution, on page 20, list a table which shows momentum is not conserved, but no reason why. I generally do not work in this area, so maybe I am missing something basic. Can anyone shed some light on this? Thanks!