Hello I have a question about sketching characteristics, I have 2 Burgers equations that are as follows: $$u_t+u\cdot u_x=0, \>\> u(x,0)=x^3$$ $$u_t+uu_x=0\>\> u(x,0)= \begin{cases} 1 & \text{ if } x\leq 0\\ 1-x & \text{ if } 0\leq x\leq 1\\ 0 & \text{ if } x\geq 1 \end{cases}$$ Now for the first I am asked to sketch the characteristics in general and sketch the profiles of $u(x,0)$ and $u(x,1)$, and for the second I am asked the sketch the characteristics in the $xt$-plane for $0\leq t\leq 1$ and then sketch $u(x,0), u(x,1/2), u(x,3/2),u(x,1)$.

I am beyond confused as to what to sketch, or even how to find the profiles.

I know the Burger's characteristics start off as vertical lines and then become less steeper until a point where the characteristics start sloping the other direction which causes shock (namely at $t=1$),and I believe the speed at which this happens is faster for the first PDE than the second, but this still doesn't help me sketch the characteristics. If anyone could help me with this I would appreciate it greatly. Thank you!

  • $\begingroup$ You can try to read in this book [Edwige Godlewski & Pierre-Arnaud Raviart] "Numerical Approximation of Hyperbolic Systems of Conservation Laws". In Example 2.1 page 13, they have your second problem. I hope it would help. It's been a long time for me since the last time I face these Burgers guys :) $\endgroup$ – JKay Sep 19 '17 at 23:52

For the first, the solution is $$u=(x-tu)^3$$ which is an implicit form to express the solution $u(x,t)$. It could be made explicit in solving the cubic equation for $u$. But it is not necessary to answer for the wanted drawing.

To draw it on the $xt$-plane, for each value of $u$, draw : $$x=u\,t+u^{1/3}$$ that is a family of straight lines.

For the second, the solution is given in PDE with strange Auxiliary Conditions

To draw it on the xt-plane, for each value of $u$, draw the corresponding function $x(t)$ derived from the equation $u(x,t)$ explicitly found.

Especially observe what occurs then $t\to 1$.


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