Riemann problem of nonconvex scalar conservation laws

Consider the scalar conservation law $\partial_t u+\partial_xf(u)=0$. Riemann problem means the initial data given by

\begin{equation} u_0=\begin{cases} u_L, & x<0 \\ u_R, & x\geq 0 \end{cases} \end{equation}

When $f(x)$ is convex, I know the corresponding theory. What if $f$ is not convex, for example $f(u)=\frac{u^3}{3}$, how to solve it?

Solution 1:

The method is very similar to the convex case, e.g. Burgers' equation where $f(u) = \frac{1}{2}u^2$, but there are more possible types of waves. In facts, in addition to shock waves and rarefaction waves, there may be waves with both discontinuous and continuous parts. Moreover, the Lax entropy condition for shocks must be replaced by the more general Oleinik entropy condition.

In the case where the flux $f$ is not convex, these are the possible types of waves:

• shock waves. If the solution is a shock wave with expression $$ u(x,t) = \left\lbrace \begin{aligned} &u_L & &\text{if}\quad x < s\, t \, ,\\ &u_R & &\text{if}\quad s\, t < x \, , \end{aligned} \right. \tag{1} $$ then the speed of shock $s$ must satisfy the Rankine-Hugoniot jump condition $s = \frac{f(u_R)- f(u_L)}{u_R - u_L}$. Moreover, the shock wave must satisfy the Oleinik entropy condition [1] $$ \frac{f(u)- f(u_L)}{u - u_L} \geq s \geq \frac{f(u_R)- f(u)}{u_R - u} , $$ for all $u$ between $u_L$ and $u_R$. In the case where $f$ is convex, the slope of its chords can be compared with its derivative using convexity inequalities. Thus, the classical Lax entropy condition $f'(u_L)>s>f'(u_R)$ is recovered, where $f'$ denotes the derivative of $f$.
• rarefaction waves. The derivation is similar to the convex case, starting with the self-similarity Ansatz $u(x,t) = v(\xi)$ where $\xi = x/t$, which gives $f'(v(\xi)) = \xi$. In the nonconvex case, the equation $f'(v(\xi)) = \xi$ may have multiple solutions $v(\xi)$, and the correct one is deduced from the continuity conditions $v(f'(u_L)) = u_L$ and $v(f'(u_R)) = u_R$. Such a solution is given by $$ u(x,t) = \left\lbrace \begin{aligned} &u_L & &\text{if}\quad x \leq f'(u_L)\, t \, ,\\ &(f')^{-1}(x/t) & &\text{if}\quad f'(u_L)\, t \leq x \leq f'(u_R)\, t \, ,\\ &u_R & &\text{if}\quad f'(u_R)\, t \leq x \, , \end{aligned} \right. \tag{2} $$ where the expression of the reciprocal $(f')^{-1}$ of $f'$ has been chosen carefully.
• compound waves, a.k.a. composite waves or semi-shocks. The latter occur when neither shock waves nor rarefaction waves are entropy solutions, but combinations of them are. The position of rarefaction parts and of discontinuous parts is deduced from the Rankine-Hugoniot condition and from the Oleinik entropy condition.

A rather practical method of solving such problems is convex hull construction: [1]

The entropy-satisfying solution to a nonconvex Riemann problem can be determined from the graph of $f (u)$ in a simple manner. If $u_R < u_L$, then construct the convex hull of the set $\lbrace (u, y) : u_R ≤ u ≤ u_L \text{ and } y ≤ f (u)\rbrace$. The convex hull is the smallest convex set containing the original set. [...] If $u_L < u_R$, then the same idea works, but we look instead at the convex hull of the set of points above the graph, $\lbrace (u, y) : u_L ≤ u ≤ u_R \text{ and } y ≥ f (u)\rbrace$.

Between $u_L$ and $u_R$, the intervals where the slope of the hull's edge is constant correspond to admissible discontinuities. The other intervals correspond to admissible rarefactions.

One can also use Osher's expression of general similarity solutions $u(x,t) = v(\xi)$, which writes [1]

$$ v(\xi) = \left\lbrace \begin{aligned} &\underset{u_L\leq u\leq u_R}{\text{argmin}} \left(f(u) - \xi u\right) && \text{if}\quad u_L\leq u_R \, ,\\ &\underset{u_R\leq u\leq u_L}{\text{argmax}} \left(f(u) - \xi u\right) && \text{if}\quad u_R\leq u_L \, . \end{aligned} \right. $$

To summarize, here are the different entropy solutions and their validity in the case $f(u) = \frac{1}{3}u^3$, where the inflection point of $f$ is located at the origin. The speed of sound is $f'(u) = u^2$, with reciprocal $(f')^{-1}(\xi) = \pm\sqrt{\xi}$. Using the convex hull construction method, one gets:

• if $[0<u_L<u_R]$ or $[u_R<u_L<0]$, the solution is a rarefaction wave $({2})$ with shape $\text{sgn}(u_R) \sqrt{x/t}$.
• else, if $[u_L<u_R< -\frac{1}{2}u_L]$ or $[-\frac{1}{2}u_L <u_R<u_L]$, the solution is a shock wave $({1})$, which speed $s = \frac{1}{3}\left( {u_L}^2 + {u_L}{u_R} + {u_R}^2 \right)$ is given by the Rankine-Hugoniot condition.
• else, if $[u_L\leq 0\leq -\frac{1}{2}u_L \leq u_R]$ or $[u_R\leq -\frac{1}{2}u_L \leq 0 \leq u_L]$, the solution is a semishock, more precisely a shock-rarefaction wave. The intermediate state $u^*$ which connects the discontinuous part to the rarefaction part satisfies $\frac{1}{3}\left( {u_L}^2 + {u_L}{u^*} + ({u^*})^2 \right) = (u^*)^2$ according to the convex hull construction, i.e. $u^* = -\frac{1}{2}u_L$. Thus, $$ u(x,t) = \left\lbrace \begin{aligned} &u_L & &\text{if}\quad x \leq \left(-{\textstyle\frac{1}{2}u_L}\right)^2\, t \, ,\\ &\text{sgn}(u_R)\sqrt{x/t} & &\text{if}\quad \left(-{\textstyle\frac{1}{2}u_L}\right)^2\, t \leq x \leq {u_R}^2\, t \, ,\\ &u_R & &\text{if}\quad {u_R}^2\, t \leq x \, . \end{aligned} \right. $$

(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.