Riemann solver

Definition

Let $\Omega \subseteq \R^m$ be open and let $F \colon \Omega \to \R^m$ be the $C^1$ flux of a hyperbolic conservation law $u_t + F(u)_x = 0$. A Riemann solver for $F$ is a map $H \colon \Omega \times \Omega \to \R^m$. Its value $H(u_l, u_r)$ is the numerical flux at an interface between a left state $u_l$ and a right state $u_r$. It is obtained from an exact or approximate solution of the Riemann problem with data $u_l, u_r$. It is required to be:

1. consistent: $H(u, u) = F(u)$ for all $u \in \Omega$;
2. locally Lipschitz continuous in both arguments.

The exact Riemann solver is $H(u_l, u_r) = F(w(0; u_l, u_r))$, where $w(x/t; u_l, u_r)$ is the self-similar entropy solution of the Riemann problem, when it exists and is unique.