An efficient operator-splitting FEM-FCT algorithm for 3D chemotaxis models
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ORIGINAL ARTICLE
An efficient operator‑splitting FEM‑FCT algorithm for 3D chemotaxis models Xueling Huang1 · Xufeng Xiao1 · Jianping Zhao1 · Xinlong Feng1 Received: 25 January 2019 / Accepted: 3 May 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract An efficient operator-splitting finite element method (FEM) combined with the flux-corrected transport (FCT) algorithm is presented to reduce the computation and storage of solving three-dimensional (3D) chemotaxis models. Firstly, the 3D coupled and positivity-preserving problem is split into a series of 1D subproblems in three spatial directions. Then each 1D subproblem is solved by the FEM-FCT algorithm which guarantees the positivity of numerical solutions. As the 1D subproblems in one direction are spatially independent, they can be calculated in parallel. Additionally, the accuracy and efficiency of the proposed method are investigated by numerical tests. Furthermore, we employ the proposed method to simulate 3D chemotaxis phenomena, including the typical blow-up effect, the more complex pattern formation and aggregating behavior of cell distribution. Keywords Chemotaxis models · Operator-splitting method · Finite element method · Flux-corrected transport algorithm Mathematics Subject Classification 65M60 · 92C15 · 92C17
1 Introduction Chemotaxis refers to the directional movement of cells or microorganisms along the concentration gradient of the chemoattractant in their tissues or living environments. This movement plays a fundamental role in the growth, development and reproduction of living things, for instance, organizing cells orientate [5], immune cells migrate [32], This work is in parts supported by the NSF of Xinjiang Province (No. 2019D01C047), the Research Fund from Key Laboratory of Xinjiang Province (No. 2017D04030), the Xinjiang Provincial University Research Foundation of China (No. XJEDU2018I002), and the NSF of China (No. 11671345, No. 11362021). * Xinlong Feng [email protected] Xueling Huang [email protected] Xufeng Xiao [email protected] Jianping Zhao [email protected] 1
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People’s Republic of China
and bacteria signal [1], etc. Therefore, it is worthwhile to research the biological chemotaxis in depth from the point of numerical simulation to reveal some chemotaxis factors and characteristics. One of the most prevalent models for chemotaxis phenomena is a series of coupled nonlinear partial differential equations, several convection–diffusion equations that control the density of cells or organisms and several reactiondiffusion equations that involve the variations of chemical concentrations [10, 13]. The generic dimensionless form of chemotaxis partial differential equations for a 3D bounded domain Ω ⊂ 𝐑3 and a time interval [0, T] reads: � � ⎧ 𝜕u ⎪ 𝜕t = 𝛼u Δu − ∇ ⋅ u𝜒(c)∇c + g(u)u, ⎪ ⎪ 𝜕c = 𝛼 Δc − 𝛽c + s(u)u, c ⎨ 𝜕t ⎪ 𝐧 ⋅ ∇u = 0, 𝐧 ⋅ ∇c = 0, ⎪ ⎪ u�t=0 = u0 , c�t=0 = c0 , ⎩
in Ω × (0, T], in Ω × (0, T], on 𝜕Ω × (0, T], in Ω,
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