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PDE Charactersitic equations colliding and shocks concepts

  1. Nov 24, 2014 #1

    I'm new to shocks and trying to get the hang of it.
    I have 3 sets of characteristic equations,( by a set I mean defined by taking a different fixed value u along the characteristic.)

    From what I understand,in general talk, we use a shock whenever two sets of characteristics collide as otherwise u=u(x,t) would be multi-valued:

    *The shock - x(t)- begins at the first collision , and the collision point acts as the initial condition for the shock curve. The shock curve remains given by the same equation until there is a change in characteristic equation to the right of the shock curve. When this occurs the new shock curve is computed subject to the u+- limits at this point where the change occurs, and this point now acting as the initial condition...

    Questions:

    For Burgers equation subject to u(x,0) = 1 for x<0, 1-x for 0<x<1, 0 for x>1.

    1) For this initial data, all curves meet at (1,1). Which set of the characteristic equations do we then treat as colliding - in attaining u+ and u-( Am I correct in thinking that u+ has to be given by u=0. but for u- don't we have a choice between u=1 and u=1-x ?)

    2) The set of characteristics between 0<s<1 converge to (1,1) and then diverge from it. Am I correct in thinking that nothing needs to be dealt with here? I.e- self-crossing characteristics is not a issue because there is no problem with u being multi-valued.

    3) For these initial conditions , I'm struggling to apply * - to the right of (1,1) . For example if I attain my shock curve at (1,1) by using u+=0 and 'choosing' u-= 1 i get the shock curve t=2x-1. This lies exactly on the characteristic given by s=1/2. That is it leaves half of this set to the left of it, so they are dealt with it. But half to right. So to the right we have these and the vertical lines given by u=0. From what I can make out at exactly (1,1) the initial condition is correct, but as soon as I begin to 'travel' along the shock t=2x-1, I can not interpret whether the change to the right of the curve is due to the line u=x-1 or u=0 (since by varying s continuously we can make either set of characteristics arbitrarily close to the shock curve).


    Thanks in advance for any assistance , extremely appreciated !!
     
  2. jcsd
  3. Nov 26, 2014 #2
    bump. please. your assistance is greatly appreciated.
     
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