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A time semi-exponentially fitted scheme for chemotaxis-growth models. (English) Zbl 1365.65201

Summary: 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.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
92C17 Cell movement (chemotaxis, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences

Software:

Chemotaxis
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