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

2nd International Conference on Mathematical Modeling in Physical Sciences 2013

物理学;数学

Athanasakis, I.E.^1 ; Papadopoulou, E.P.^1 ; Saridakis, Y.G.^1

Applied Math and Computers Lab, Technical University of Crete, 73100 Chania, Crete, Greece^1

关键词: Convergence and stability;    High-order numerical method;    Nonlinear reaction-diffusion equations;    Numerical experimentations;    Spatial discretization methods;    Spatio-temporal dynamics;    Strong stability preserving;    Temporal discretization;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/490/1/012133/pdf
DOI  :  10.1088/1742-6596/490/1/012133

来源: IOP

【 摘 要 】

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.