On a linear DG approximation of chemotaxis models with damping gradient nonlinearities

TL;DR

This paper introduces a new linear positivity-preserving discontinuous Galerkin method for chemotaxis models with damping gradient nonlinearities, effectively preventing blow-up in chemotactic collapse scenarios.

Contribution

The work develops a novel linear DG approximation that preserves positivity and handles complex chemotaxis models with damping nonlinearities, including local and nonlocal cases.

Findings

Numerical experiments confirm the method's effectiveness in preventing blow-up.

The approach aligns with previous analysis and demonstrates stability.

Damping gradient terms can control chemotactic collapse.

Abstract

In this work we present a novel linear and positivity preserving upwind discontinuous Galerkin (DG) approximation of a class of chemotaxis models with damping gradient nonlinearities. In particular, both a local and a nonlocal model including nonlinear diffusion, chemoattraction, chemorepulsion and logistic growth are considered. Some numerical experiments in the context of chemotactic collapse are presented, whose results are in accordance with the previous analysis of the approximation and show how the blow-up can be prevented by means of the damping gradient term.