Forgive me for the long post, but I'm in some desperate need of clarity on this matter. I just can't seem to grasp the whole shock wave concept, or at least the meaty part of it . I only have a couple of problems left to do to finish my HW I'm at an impasse until I dispel my confusion. I really hope someone here can help clear my confusion.(adsbygoogle = window.adsbygoogle || []).push({});

With the PDE we're working with (Inviscid Burgers' eqn u u_x + u_t = 0) the characteristics are lines and u(x, y) is constant along them. What I don't quite get is what happens exactly when a discontinuity is introduced and a shock wave develops.

Suppose we start at some x0 for which u(x0, 0) = ul is constant along some characteristic. At some (x*, t*), u(x, y) would be multi-valued and thus a shock develops. It makes sense to me that no value of u exists on the shock line and that physically u still propogates because of conservation laws. What I can't seem to grasp is how we find the value ur after the shock and what characteristic it is governed by.

I know that given the conservation law between rate of change of u and flux across the discontinuity (s-[q] = 0 where [x] defines xl - xr and s is the shock speed) we can find the jump conditions, but I don't understand what those mean or how to phrase them exactly. Also, if ul jumps to ur, what characteristic does it jump to? What confuses me is that other characteristics eventually collide as well for later times even though physically there may have only been one discontinuity. Do we no longer look to characteristics after the discontinuity? Maybe we ignore further collisions since we crossed the discontinuity already?

Very much thanks to anyone that can help me understand this better or point me somewhere I can find out. My textbooks suck and my prof is out of town.

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# PDEs and shock waves

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