The Rankine-Hugoniot condition

Question

In this notes, example 5, page 5. It gives an example on how to solve a PDE where characteristics might cross using the Rankine-Hugoniot condition. The example is in an initial-value problem for Burgers' equation $u_t + u u_x = 0$, with piecewise linear initial data $$ u(x,0) = \phi(x) = \left\lbrace\begin{aligned} &1 &&\text{if}\quad x<0\\ &1-x &&\text{if}\quad 0<x<1\\ &0 &&\text{if}\quad 1<x \end{aligned}\right. $$ I have two questions about it:

1). In my opinion, Rankine-Hugoniot condition is just a necessary condition for the discontinuous points of a solution. How does it guarantee the thing we get is indeed a solution?

2).Using the Rankine-Hugoniot condition we can only calculate the speed of the curve of discontinuities. How do we locate its exact formula?(i.e. how to choose one-point on it). Can we choose any arbitrary point where the characteristics cross?

Answer 1

If needed, here is a sketch of the characteristic curves in $x$-$t$ plane, which may help to see better what happens:

The solution obtained by the method of characteristics up to the breaking time satisfies $u = \phi (x-ut)$ for $t<1$, i.e. $$ u(x,t) = \left\lbrace\begin{aligned} &1 & &\text{if}\quad x \leq t \\ &\tfrac{1-x}{1-t} & &\text{if}\quad t \leq x\leq 1\\ &0 & &\text{if}\quad x\geq 1 \end{aligned}\right. $$ At the breaking time, the shock speed $s$ is given by the Rankine-Hugoniot condition $s = \tfrac12 (1 + 0)$. Therefore, after the breaking time $t\geq 1$, $$ u(x,t) = \left\lbrace\begin{aligned} &1 & &\text{if}\quad x < 1 +\tfrac12 (t-1) \\ &0 & &\text{if}\quad x > 1 +\tfrac12 (t-1) \end{aligned}\right. $$

To your questions:

1. The Rankine-Hugoniot condition $[\![\frac{1}{2}u^2]\!] = s [\![u]\!]$ is a condition satisfied by discontinuous weak solutions of the Burgers' equation. A weak solution which satisfies the Rankine-Hugoniot condition is not necessarily a solution to the proposed initial-value problem $u(x,0) = \phi(x)$. To be a solution, it must be in agreement with the characteristics equation until they cross, i.e. the fact that $u$ is constant along the curves $u = \phi(x - ut)$. Moreover, since weak solutions are not unique, it must satisfy an additional entropy condition. In the case of Burgers' equation, the Lax entropy condition amounts to the following statement: "if characteristics cross, then a shock-wave arises; but if they separate, a rarefaction wave occurs".
2. The location of a discontinuity is deduced from the characteristics equation, whereas its speed is deduced from the Rankine-Hugoniot condition. In the present case, a shock occurs at the time $t=1$ (characteristics intersect). Its speed deduced from the Rankine-Hugoniot condition is $s=\frac{1}{2}$.