Rarefaction colliding with two shocks

The Burgers equation {v_t+vv_x =0} is a great illustration of shock formation and propagation, with {v(x,t)} representing velocity at time {t} at position {x}. Let’s consider it with piecewise constant initial condition

\displaystyle u(x,0) = \begin{cases} 1,\quad & x<0 \\ 0,\quad & 0<x<1 \\ 2,\quad & 1<x<2 \\ 0,\quad & x>2 \end{cases}

The equation, rewritten as {(v_t,v_x)\cdot (1,v)=0}, states that the function {v} stays constant on each line of the form {x= x_0 + v(x_0,0)t}. Since we know {v(x_0,0)}, we can go ahead and plot these lines (characteristics). The vertical axis is {t}, horizontal {x}.

burger0

Clearly, this picture isn’t complete. The gap near {1} is due to the jump of velocity from {0} to {2}: the particles in front move faster than those in back. This creates a rarefaction wave: the gap gets filled by characteristics emanating from {(1,0)}. They have different slopes, and so the velocity {v} also varies within the rarefaction: it is {v(x,t)=(x-1)/t}, namely the velocity with which one has to travel from {(1,0)} to {(x,t)}.

burger1

The intersecting families of characteristics indicate a shock wave. Its propagation speed is the average of two values of {v} to the left and to the right. Let’s draw the shock waves in red.

burger2

Characteristics terminate when they run into shock. Truncating the constant-speed characteristics clears up the picture:

burger3

This is already accurate up to the time {t=1}. But after that we encounter a complication: the shock wave to the right separates velocity fields of varying intensity, due to rarefaction to the left of the shock. Its propagation is now described by the ODE

\displaystyle \frac{dx}{dt} = \frac12 \left( \frac{x-1}{t} + 0 \right) = \frac{x-1}{2t}

which can be solved easily: {x(t) = 2\sqrt{t} +1}.

Similarly, the shock on the left catches up with rarefaction at {t=2}, and its position after that is found from the ODE

\displaystyle \frac{dx}{dt} = \frac12 \left( 1 + \frac{x-1}{t} \right) = \frac{x-1+t}{2t}

Namely, {x(t) = t-\sqrt{2t}+1} for {t>2}. Let’s plot:

burger4

The space-time trajectories of both shocks became curved. Although initially, the shock on the right moved twice as fast as the shock on the left, the rarefaction wave pulls them toward each other. What is this leading to? Yet another collision.

The final collision occurs at the time {t=6+4\sqrt{2} \approx 11.5} when the two shock waves meet. At this moment, rarefaction ceases to exist. The single shock wave forms, separating two constant velocity fields (velocities {1} and {0}). It propagates to the right at constant speed {1/2}. Here is the complete picture of the process:

burger5

I don’t know where the holes in the spacetime came from; they aren’t predicted by the Burgers equation.

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