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 equationsResumo
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.
