An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems
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Department of Mathematics, Michigan State University, East Lansing, MI 48864
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2.
Department of Applied Mathematics, Providence University, Taichung
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Received:
01 July 2005
Accepted:
29 June 2018
Published:
01 February 2007
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MSC :
65N06, 65N12, 65N55, 92-08.
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Reaction-diffusion-chemotaxis systems have proven to be fairly
accurate mathematical models for many pattern formation problems in chemistry
and biology. These systems are important for computer simulations
of patterns, parameter estimations as well as analysis of the biological systems.
To solve reaction-diffusion-chemotaxis systems, efficient and reliable
numerical algorithms are essential for pattern generations. In this paper, a
general reaction-diffusion-chemotaxis system is considered for specific numerical
issues of pattern simulations. We propose a fully explicit discretization
combined with a variable optimal time step strategy for solving the reactiondiffusion-
chemotaxis system. Theorems about stability and convergence of the
algorithm are given to show that the algorithm is highly stable and efficient.
Numerical experiment results on a model problem are given for comparison
with other numerical methods. Simulations on two real biological experiments
will also be shown.
Citation: Chichia Chiu, Jui-Ling Yu. An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 187-203. doi: 10.3934/mbe.2007.4.187
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Abstract
Reaction-diffusion-chemotaxis systems have proven to be fairly
accurate mathematical models for many pattern formation problems in chemistry
and biology. These systems are important for computer simulations
of patterns, parameter estimations as well as analysis of the biological systems.
To solve reaction-diffusion-chemotaxis systems, efficient and reliable
numerical algorithms are essential for pattern generations. In this paper, a
general reaction-diffusion-chemotaxis system is considered for specific numerical
issues of pattern simulations. We propose a fully explicit discretization
combined with a variable optimal time step strategy for solving the reactiondiffusion-
chemotaxis system. Theorems about stability and convergence of the
algorithm are given to show that the algorithm is highly stable and efficient.
Numerical experiment results on a model problem are given for comparison
with other numerical methods. Simulations on two real biological experiments
will also be shown.
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