Riemann problem

Remarks

The equation and the data are unchanged by the scaling $(x, t) \mapsto (\lambda x, \lambda t)$ for $\lambda > 0$, so one looks for self-similar solutions $u(x, t) = v(x / t)$.

Scalar case with convex flux. Let $m = 1$ and let $F$ be smooth and uniformly convex, and let $G$ be the inverse function of $F'$. If $u_l > u_r$, the entropy solution is the shock wave equal to $u_l$ for $x < \sigma t$ and to $u_r$ for $x > \sigma t$, where $\sigma = \dfrac{F(u_l) - F(u_r)}{u_l - u_r}$. If $u_l < u_r$, the entropy solution is the rarefaction wave equal to $u_l$ for $x < F'(u_l) t$, to $G(x/t)$ for $F'(u_l) t \le x \le F'(u_r) t$, and to $u_r$ for $x > F'(u_r) t$.

Systems. Consider a strictly hyperbolic system whose characteristic fields are each genuinely nonlinear or linearly degenerate. Lax showed that its Riemann problem has a unique solution made of at most $m$ elementary waves when $\abs{u_l - u_r}$ is small. These waves are shocks, rarefaction waves and contact discontinuities, one for each characteristic family, separated by constant states. For large data, or for more general systems, solutions may fail to exist or to be unique in this class. For the Euler equations of gas dynamics, the problem posed by Riemann, the solution is known explicitly up to solving one scalar equation, and it may contain a vacuum.

Solutions of Riemann problems are the building blocks of Godunov's scheme and of the Glimm scheme.