Convergence of a time discrete scheme for a chemotaxis-consumption model

TL;DR

This paper introduces a new time-discrete numerical scheme for a chemotaxis-consumption model, proving its convergence to weak solutions and providing a stable computational approach for simulating cell and chemical dynamics.

Contribution

The paper develops a novel scheme using variable reformulation and truncation, with proofs of existence, stability, and convergence for the chemotaxis-consumption system.

Findings

The scheme converges to a weak solution of the model.

Uniform a priori estimates are established.

Two methods for approximating the chemical concentration are proposed.

Abstract

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.