# Numerical Resolution of the One-dimensional Shallow Water Equations by the Discontinuous Galerkin Method

## Palavras-chave:

Finite Element Method, Hyperbolic Systems, Shallow water equations, Discontinuous Galerkin, Saint- Venant equations## Resumo

The 1D shallow water equations model the unstable flow of an incompressible newtonian fluid in a

channel. These equations are given by the conservation of mass and linear momentum, and can be viewed as an

average of the Navier-Stokes equations under the assumption that the vertical length scale is much smaller than the

horizontal length scale. These equations, also known as Saint-Venant equations, compose a system of conservation

laws of hyperbolic nature, allowing for discontinuous solutions. In order to provide a numerical methodology for

the approximation of the Saint-Venant equations capable of accurately representing such discontinuous solutions,

in this work we adopt the Discontinuous Galerkin method. This class of finite element methods is based on

the weak formulation of the differential equation to be studied, and uses discontinuous piecewise polynomial

approximations. We present some numerical experiments for different flow regimes and non-horizontal beds for

the 1D shallow water problem, such as idealized dam breaking, hydraulic jumps and transcritical flows in channels

with non-constant bathymetry.