A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

Jianping Lv; Chunguang Li; Jianqiang Dong; Jianping Lv; Chunguang Li; Jianqiang Dong

doi:10.3934/math.2024189

AIMS Mathematics

2024, Volume 9, Issue 2: 3836-3849. doi: 10.3934/math.2024189

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Research article

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China

Received: 19 November 2023 Revised: 27 December 2023 Accepted: 08 January 2024 Published: 11 January 2024
MSC : 35A35, 35B20, 35C20

In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.
- plant water model for diffusion,
- high accuracy compact difference scheme,
- Taylor series expansion,
- numerical simulation
Citation: Jianping Lv, Chunguang Li, Jianqiang Dong. A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment[J]. AIMS Mathematics, 2024, 9(2): 3836-3849. doi: 10.3934/math.2024189

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Abstract

In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.

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