In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any $s\ge 1$), $$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v \quad \hbox{in $(0,T)\times \Omega$,}$$ endowed with isolated boundary conditions and initial conditions, where $(u,v)$ model cell density and chemical signal concentration. The proposed scheme is defined via a reformulation of the model, using the auxiliary variable $z = \sqrt{v + \alpha^2}$ combined with a Backward Euler scheme for the $(u,z)$-problem and a upper truncation of $u$ in the nonlinear chemotaxis and consumption terms. Then, two different ways of retrieving an approximation for the function $v$ are provided. We prove the existence of solution to the time discrete scheme and establish uniform in time \emph{a priori} estimates, yielding the convergence of the scheme towards a weak solution $(u,v)$ of the chemotaxis-consumption model.

翻译：本文针对以下趋化-消耗模型（其中$s\ge 1$）提出并研究了一种时间离散格式：在$(0,T)\times \Omega$上，方程$\partial_t u - \Delta u = - \nabla \cdot (u \nabla v)$和$\partial_t v - \Delta v = - u^s v$配备孤立边界条件和初始条件，其中$(u,v)$分别表示细胞密度和化学信号浓度。该格式基于模型重构，引入辅助变量$z = \sqrt{v + \alpha^2}$，结合针对$(u,z)$问题的向后欧拉格式，并对非线性趋化和消耗项中的$u$进行上截断。随后，提供了两种恢复函数$v$近似值的不同方法。我们证明了该时间离散格式解的存在性，并建立了关于时间的一致先验估计，从而证明了该格式收敛于趋化-消耗模型的一个弱解$(u,v)$。