# Why does burgers equation become discontinuous over time?

We have burgers equation $$U_t+UU_x=0$$.

We set the following initial conditions:

• $$U(x,0) = 1 + 2/\pi \cdot \arctan(x)$$

with clamped endpoints

• $$U_\text{right endpoint}=0$$
• $$U_\text{left endpoint}=2$$

Solving using the finite difference method with Forward Euler gives the following plot over time.

We have that the solution becomes 'square' and/or discontinuous over time. I am wondering what could explain this behavior. I believe it has to do with the top of the wave traveling faster than the bottom of the wave. However, this seems unsatisfactory.

• Hi! I hope the pages 25-27 from Numerical Method for Conservation Laws, by R.J. LeVeque, will help you. The pages study the formation of shock at Burger's Equation. The shock occur when $U_0'(x)$ is negative at same point. It has relation with the inclination of carachteristics. Good studies. – Quiet_waters Dec 6 '19 at 0:52
• @Na'omi Thanks. That is a very helpful resource. Here is the link for those that come along this post pdfs.semanticscholar.org/1470/… – Jac Frall Dec 6 '19 at 1:17
• You're welcome. – Quiet_waters Dec 6 '19 at 13:01

You may have a look at this post, where the computation of the breaking time is presented. Actually, it is easy to get an answer based on intuition. Indeed, the traveling speed of Burgers' equation $$U_t+UU_x=0$$ is $$U$$. Since the data is zero on the right side, nothing moves there. However, the data on the left side is $$U=2$$, and it travels towards increasing $$x$$ at that speed. Therefore the left part is faster than the right part, and the waveform will break.