Semi-implicit finite volume schemes for a chemotaxis-growth model

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• 作者：M. Akhmouch ^{mohammed.akhmouch@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; M. Benzakour Amine ; ^{mohammed.benzakour2@usmba.ac.ma" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Finite volume scheme ; Semi-implicit scheme ; Convergence analysis ; Chemotaxis ; Pattern formation ; Growth
• 刊名：Indagationes Mathematicae
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：27
• 期：3
• 页码：702-720
• 全文大小：1404 K

文摘

This paper is concerned with finite volume approximations for a nonlinear parabolic–elliptic system for chemotaxis-growth in R^d${\mathbb{R}}^d$, d=2,3$d=2,3$. This model describes a process of pattern formation by some chemotactic biological individuals. We present two schemes which make use of a semi-implicit time discretization and an upwind finite volume approximation. For both schemes, we prove existence, uniqueness and nonnegativity of the approximate solutions under some conditions on the time step, and we show (for one of the schemes) that the numerical solution converges to a corresponding weak solution for the studied model. Numerical simulations are performed in two dimensional spaces to demonstrate the efficiency of the schemes to capture the pattern formations and to verify our theoretical results.