# Shallow Water Equations

## Model Data

**Type:**
classic pde
fluid dynamics

**Physics Modes:**
custom equation
convection and diffusion

**Keywords:**
shallow water equation
equation editing

Saint-Venant shallow water equations is a simplified model of fluid flow with a free surface. The non-conservative form of the equations read

\[
\left\{\begin{array}{;‘}
\ \frac{\partial h}{\partial t} + (u\frac{\partial h}{\partial x} + v\frac{\partial h}{\partial y} ) + (h+H)(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) = 0 \\\

\frac{\partial u}{\partial t} + (u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} ) = -g\frac{\partial h}{\partial x} \\\

\frac{\partial v}{\partial t} + (u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} ) = -g\frac{\partial h}{\partial y}
\end{array}\right.
\]

where *h* is the unknown free surface height relative to the mean level *H*.

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