  The well-suited discretization of the Keller-Segel equations for chemotaxis
has become a very challenging problem due to the convective nature inherent to
them. This paper aims to introduce a new upwind, mass-conservative, positive
and energy-dissipative discontinuous Galerkin scheme for the Keller-Segel
model. This approach is based on the gradient-flow structure of the equations.
In addition, we show some numerical experiments in accordance with the
aforementioned properties of the discretization. The numerical results obtained
emphasize the really good behaviour of the approximation in the case of
chemotactic collapse, where very steep gradients appear.
