CONVERGENCE OF A TIME DISCRETE SCHEME FOR A CHEMOTAXIS-CONSUMPTION MODEL

In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any s \geq 1): \partial tu - \Delta u = -V \cdot (uVv), \partial tv - \Delta v = -usv 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 = v + \alpha 2 combined with a backward Euler scheme for the (u, z)-problem and an 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 a priori estimates, yielding the convergence of the scheme towards a weak solution (u, v) of the chemotaxis-consumption model.