In this work, we prove the convergence of a discretized anisotropic degenerate parabolic system modeling chemotaxis. The used method belongs to a positive nonlinear Discrete Duality Finite Volume (DDFV) family. Its aim is to respect the physical bounds on the computed solution independently of the used mesh and the involved anisotropy. To achieve that, two main terms referred to as convection and diffusion are carefully approximated. The transport contribution is discretized using a product of centered and upstream schemes. The discretization of the oscillatory diffusive fluxes is treated by means of an upwind scheme. Based on the established a priori estimates, we demonstrate the convergence of the approximate solutions thanks to compactness arguments adapted to degenerate parabolic problems. Numerical experiments are performed in two dimensions. The goal is to test the efficiency and the robustness of the proposed scheme on a series of examples including anisotropic tensors and non-smooth data over various types of meshes.

在这项工作中，我们证明了建模趋化性的离散各向异性简并抛物线系统的收敛性。使用的方法属于正非线性离散对偶有限体积（DDFV）系​​列。其目的是独立于所使用的网格和所涉及的各向异性而尊重计算解上的物理边界。为此，我们仔细地近似了两个对流和扩散这两个主要术语。使用集中计划和上游计划的乘积来离散化交通贡献。振荡扩散通量的离散化通过迎风方案进行处理。基于已建立的先验估计，我们证明了近似解的收敛性，这要归功于适用于退化抛物线问题的紧致性参数。数值实验在二维上进行。