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.

翻译：趋化性Keller-Segel方程因其固有的对流特性，其离散化处理已成为极具挑战性的问题。本文旨在为Keller-Segel模型引入一种新的迎风、质量守恒、正性与能量耗散间断Galerkin格式。该方法基于方程的梯度流结构。此外，我们展示了与上述离散特性一致的一些数值实验。获得的数值结果强调了该近似在趋化性塌缩情况下的优异表现，其中会出现非常陡峭的梯度。