These equations are widely used in meteorology og oceanography to model phenomena such as tsunamis and dam breaches. The system takes the form \begin{align*} h_t + (uh)_x &= 0,\\ (uh)_t + (hu^2 + \frac 12gh^2)_x &= 0, \end{align*}

Google: Shallow water equations, hyperbolic conservation laws.

Challenges: nonlinearity, applications