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

Daniel Acosta-Soba, Alessandro Columbu, J. Rafael Rodríguez-Galván

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.

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

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.
Paper Structure (12 sections, 8 theorems, 38 equations, 7 figures)

This paper contains 12 sections, 8 theorems, 38 equations, 7 figures.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Chemotaxis models with damping gradient nonlinearities
  4. Existence and regularity of solutions
  5. Numerical approximation
  6. Notation
  7. Fully discrete scheme
  8. Properties of the scheme
  9. Numerical experiments
  10. Only attraction local model
  11. Attraction-repulsion local model
  12. Attraction-repulsion nonlocal model

Key Result

Proposition 3.2

Let $\mu>0$ and $1\le\rho<k$. The mass of the cell density $u$ in problem:chemotaxis_local or problem:chemotaxis_nonlocal is bounded as follows: where $T$ is the upper bound of the time interval where the function $u$ is defined, with $T\le\infty$.

Figures (7)

  • Figure 1: $\Pi^h_1 u^m$ at different time steps in Test \ref{['sec:test_1']} ($c=0$, $\chi=5$, $\xi=0$)
  • Figure 2: $v^{m}$ at different time steps in Test \ref{['sec:test_1']} ($c=0$, $\chi=5$, $\xi=0$)
  • Figure 3: $\|u ^m\|_{L^\infty(\Omega)}$ over time for different values of $c$ and $\gamma$ in Test \ref{['sec:test_1']} ($\chi = 5$, $\xi = 0$)
  • Figure 4: $\|u ^m\|_{L^\infty(\Omega)}$ over time for different values of $c$ and $\gamma$ in Test \ref{['sec:test_2']} ($\chi = 5$, $\xi = 1$)
  • Figure 5: $\Pi^h_1 u^{m}$ at different time steps in Test \ref{['sec:test_3']} ($c=0$, $\alpha=1.5$)
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.2
  • Theorem 3.3: Local classical solution of \ref{['problem:chemotaxis_local']} and \ref{['problem:chemotaxis_nonlocal']}
  • Theorem 3.4: Global and bounded classical solution of \ref{['problem:chemotaxis_local']}
  • Theorem 3.5: Global and bounded classical solution of \ref{['problem:chemotaxis_nonlocal']}
  • Remark 3.6
  • Lemma 4.1: Local mass bounds
  • proof
  • ...and 7 more