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I am studying conservation laws and hyperbolic systems, particularly, Burgers' equation and shocks, and have a doubt at pages 48/49 of the book Numerical Methods for Conservation Laws by R.J. LeVeque (Birkäuser, 1992).

Sometimes, the characteristics' method give us some triple-valued solution to a question, particularly, a Riemann problem, what seems like a graphic with a bulge, like this

image of graphic triple-valued

At page 48, the author says:

Note that the characteristic velocities are $f'(u)$ so that the profile of this bulge seen here at time $t$ is simply the graph of $tf'(u)$ turned sideways.

I could not understand this. Note that this doubt, if I understood all right, is essencial to use the equal area rule. Is this valid always? Why?

Many thanks.

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The same content is exposed in a section of Ref. (1) on the Buckley-Leverett equation as follows (p. 352-353):

We only need to solve for $q$, the water saturation, and so we can solve a scalar conservation law $$ q_t + f(q)_x = 0 $$ with the flux $f(q)$ given by [ $$ f(q) = \frac{q^2}{q^2 + a (1-q)^2} . \tag{16.2} $$ Here $a<1$ is a constant]. Note that this flux is nonconvex, with a single inflection point.

and

Now consider the Riemann problem with initial states $q_l = 1$ and $q_r = 0$. By following characteristics, we can construct the triple-valued solution shown in [Fig. 4.7(a) of OP]. Note that the characteristic velocities are $f'(q)$, so that the profile of this bulge, seen here at time $t$, is simply the graph of $t f'(q)$ turned sideways.

Thus, we need to solve a Riemann problem in the nonconvex case. We consider a self-similar solution of the form $q(x,t) = v(\xi)$ with $\xi = x/t$. Using the chain rule, we have $q_t = v'(\xi)\xi_t$ and $q_x = v'(\xi)\xi_x$ with $\xi_t = -\xi/t$ and $\xi_x = 1/t$. Injecting the previous Ansatz in the quasi-linear PDE $q_t + f'(q)q_x = 0$ satisfied by the water saturation leads to $$ \left(f'(v(\xi)) - \xi\right) {v'(\xi)} = 0\, , $$ which nontrivial solutions satisfy $f'(q(x,t)) = x/t$. Hence, the bulge at some time $t$ is obtained by plotting $x = t f'(q)$ for $q$ in $[0,1]$. This is illustrated below, where $a=1/2$ and $t=1$ (abscissas $q$, ordinates $x$):

bulge

It remains to rotate the figure to obtain $q$ as a function of $x$. This topic is also well-explained in an interactive book, more precisely §1.6 and §5.1 on nonconvex conservation laws.

The equal area rule results from conservation, more precisely from the fact that a discontinuous weak solution must have the same integral as the smooth multivalued solution deduced from the characteristics. This is exactly the way it is applied in Fig. 4.7(b) of OP.


(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002, doi:10.1017/CBO9780511791253

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  • $\begingroup$ Hi, many thanks...! I am studiyng your answer. $\endgroup$ – Na'omi May 11 at 15:35
  • $\begingroup$ I am afraid that I could not understand very well until now: I am OK until "Injecting this Ansatz...". However, I could not get yet why I can jump from this to "Hence, the bulge at some time...". I could not see the relation between these two phrases... Many thanks. $\endgroup$ – Na'omi May 12 at 18:15
  • $\begingroup$ I think I am understanding a little bit more, even though I have difficulties. I thank you very much and hope you can continue to follow my questions on modeling math. $\endgroup$ – Na'omi May 14 at 17:05

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