Study of a chemo-repulsion model with quadratic production. Part I: Analysis of the continuous problem and time-discrete numerical schemes

TL;DR

This paper analyzes a chemo-repulsion model with quadratic production, establishing existence, regularity, and stability of solutions, and introduces two numerical schemes with proven convergence and error estimates, supported by simulations.

Contribution

It provides the first comprehensive analysis of continuous solutions and unconditionally energy-stable numerical schemes for the chemo-repulsion model with quadratic production.

Findings

Global weak solutions are obtained for the model.

Two energy-stable time schemes are developed and analyzed.

Numerical simulations illustrate the schemes' behavior.

Abstract

We consider a chemo-repulsion model with quadratic production in a bounded domain. Firstly, we obtain global in time weak solutions, and give a regularity criterion (which is satisfied for $1D$ and $2D$ domains) to deduce uniqueness and global regularity. After, we study two cell-conservative and unconditionally energy-stable first-order time schemes: a (nonlinear and positive) Backward Euler scheme and a linearized coupled version, proving solvability, convergence towards weak solutions and error estimates. In particular, the linear scheme does not preserve positivity and the uniqueness of the nonlinear scheme is proved assuming small time step with respect to a strong norm of the discrete solution. This hypothesis is reduced to small time step in $nD$ domains ($n\le 2$) where global in time strong estimates are proved. Finally, we show the behavior of the schemes through some numerical simulations.