Triple-valued solution to Riemann problem – Profile of the bulge

Question

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

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.

Answer 1

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 $x$):

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.

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