, Volume 54, Issue 2, pp 609–641 | Cite as

A time semi-exponentially fitted scheme for chemotaxis-growth models

  • M. Akhmouch
  • M. Benzakour AmineEmail author


In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The numerical approximation makes use of a standard finite volume scheme in space with a special treatment for the convection-diffusion fluxes which are approximated by the classical Il’in fluxes. For the time discretization, we introduce our linearized semi-exponentially fitted scheme. The paper gives a comparison between the proposed scheme and different versions of linearized backward Euler schemes. The existence and uniqueness of a numerical solution to the scheme and its convergence to a weak solution of the studied system are proved. In the last section, we present some numerical tests to show the performance of our method. Our numerical approach is then applied to a chemotaxis-growth model describing bacterial pattern formation.


Chemotaxis-growth Finite volume method Time discretization Exponential fitting 

Mathematics Subject Classification

65M08 65M12 92C17 



The authors would like to thank the anonymous referees for their patience and for their many constructive comments.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human participants or animals

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


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Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Département de Mathématiques, Faculté des sciences et techniques de FèsUSMBAFèsMorocco

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