Inviscid Burgers' equation: drawing the shock

Question

Solve Burgers' equation $$ \left\{\begin{aligned} u_{t}+uu_x &=0 \quad \text { for } \quad t>0 \\ u(x, 0) &=u_{0}(x) \end{aligned}\right. $$ with $u=u(x,t)$ and the side condition $u(x,0)=-x$.

I am aware that a similar question with initial condition u=x has been asked before, and asked this because I was wondering what the difference would be when the characteristic lines are set to converge.

Ninad Munshi · Accepted Answer · 2020-12-01T15:34:52.270

3

From implicit function theorem we have the following

$$u_t+uu_x = 0 \implies \frac{dx}{dt}=u$$

So in other words, the slopes of the characteristics depend on the value of $u$. With $u=x$, you can see that the characteristics that start at negative $x$ move left (negative slope) and vice versa for positive $x$. Could you reason out the behavior for $u=-x$ instead?

Bonus question, technically the shock in both situations could be chosen to be anything, but how does one pick out the max entropy solution for both $u=x$ and $u=-x$ ?

edited Dec 01 '20 at 15:34

answered Dec 01 '20 at 05:24

Ninad Munshi

34,407

1

Did you mean to write $u = dx/dt$? – Sally G Dec 01 '20 at 15:31
@SallyG You're absolutely right! Let me fix that – Ninad Munshi Dec 01 '20 at 15:34
Thanks, is my guess correct, too? – Sally G Dec 01 '20 at 15:34
@SallyG Interesting, I usually don't solve the Burgers equation explicitly but that solution checks out. What happens to the characteristics when they intersect $x=0$ ? – Ninad Munshi Dec 01 '20 at 15:36
I'm new to this topic, so I'm not sure. Maybe this is where the sonic-wave-shock happens? – Sally G Dec 01 '20 at 15:38
For the max entropy solution (the physical one where all the characteristics move at the same time) the characteristics will terminate at the $x=0$ line. That is called a shock. – Ninad Munshi Dec 01 '20 at 15:41
I see. Learned a lot of vocab today. Thanks – Sally G Dec 01 '20 at 15:44
Let us continue this discussion in chat. – Sally G Dec 01 '20 at 15:45

Inviscid Burgers' equation: drawing the shock

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