HLL two-wave approximate flux

For a two-state problem for a flux $f$ on the open state domain $G$ of Strictly hyperbolic conservation systems and characteristic families, choose explicit speeds $S_L<S_R$. A signal speed means the propagation speed of one wave in the particular Riemann solution being approximated. If the HLL model is used as an enclosing approximation to that fan, its admissibility hypothesis is that every such speed lies in $[S_L,S_R]$; this definition supplies no general wave-speed bound. Set $f_L=f(u_L)$ and $f_R=f(u_R)$. The HLL flux is ${\widehat{f}}_{\mathrm{H}\mathrm{L}\mathrm{L}}=\{\begin{array}{ll}{f}_L,& 0\le S_L,\\ {\displaystyle \frac{S_R{f}_L-S_L{f}_R+S_L{S}_R(u_R-u_L)}{S_R-S_L}},& S_L<0<S_R,\\ f_R,& S_R\le 0.\end{array}$ In the middle case the model has three constant regions separated by the two waves $S_L,S_R$ and one intermediate state. The wave-speed enclosure is a separate hypothesis whenever an approximation claim uses it.