Shallow water equations

Remarks

The system has the form $\partial_t U + \partial_x \Phi(U) = 0$ for $U = (h, hu)$, so it is a system of hyperbolic conservation laws. The eigenvalues of the Jacobian matrix of $\Phi$ are $u \pm \sqrt{g h}$, which are real and distinct when $h > 0$.

Example: for constants $H > 0$ and $U_0 \in \R$, the uniform flow $h = H$, $u = U_0$ is a solution, since all derivatives vanish.

Non-example: $h(x, t) = 1 + x^2$ and $u = 0$ do not form a solution, because $\partial_x(\tfrac{1}{2} g h^2) = g h \, \partial_x h = 2 g x (1 + x^2)$ is not identically $0$.

In two dimensions, with velocity $(u, v)$, the equations are $\partial_t h + \nabla \cdot (h u, h v) = 0$ together with two momentum equations of the same form. Variants add a varying bottom, friction or the Coriolis force.