Runge-Kutta and Hermite Collocation for a biological invasion problem modeled by a generalized Fisher equation

Fisher's equation has been widely used to model the biological invasion of single-species communities in homogeneous one dimensional habitats. In this study we develop high order numerical methods to accurately capture the spatiotemporal dynamics of the generalized Fisher equation, a nonlinear...

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Bibliographic Details
Published in:Journal of physics. Conference series 2014-01, Vol.490 (1), p.12133-4
Main Authors: Athanasakis, I E, Papadopoulou, E P, Saridakis, Y G
Format: Article
Language:English
Subjects:
Biological
Biological models (mathematics)
Collocation
Discretization
Experimentation
Mathematical analysis
Mathematical models
Nonlinear equations
Nonlinearity
Numerical methods
Physics
Reaction-diffusion equations
Runge-Kutta method
Stability
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Summary:Fisher's equation has been widely used to model the biological invasion of single-species communities in homogeneous one dimensional habitats. In this study we develop high order numerical methods to accurately capture the spatiotemporal dynamics of the generalized Fisher equation, a nonlinear reaction-diffusion equation characterized by density dependent non-linear diffusion. Working towards this direction we consider strong stability preserving Runge-Kutta (RK) temporal discretization schemes coupled with the Hermite cubic Collocation (HC) spatial discretization method. We investigate their convergence and stability properties to reveal efficient HC-RK pairs for the numerical treatment of the generalized Fisher equation. The Hadamard product is used to characterize the collocation discretized non linear equation terms as a first step for the treatment of generalized systems of relevant equations. Numerical experimentation is included to demonstrate the performance of the methods.
ISSN:1742-6596
1742-6588
1742-6596
DOI:10.1088/1742-6596/490/1/012133