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Euler Equations, Sod shock tube & conservation

  1. Jun 27, 2014 #1
    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
    [tex]
    \begin{bmatrix}
    \rho \\ \rho u \\ E
    \end{bmatrix}_t
    +
    \begin{bmatrix}
    \rho u \\ \rho u^2 + p \\ u(E +p)
    \end{bmatrix}_x
    = 0
    [/tex]
    where pressure is related to the conserved quantities by
    [tex]
    p = (\gamma - 1)(E - 0.5 \rho u^2)
    [/tex]
    The Sod shock problem splits the domain into two regions separated by a density and pressure discontinuity with initial velocity zero. That is, [tex]\rho_L = 1, \rho_R = .125; p_L=1, p_R =.1; u_L=u_R =0 [/tex] 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!
     
  2. jcsd
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