Numerical simulation of high-dimensional two-component reaction–diffusion systems with fractional derivatives

ABSTRACT

Two-component reaction–diffusion models in high dimensions involving space-fractional derivatives are used as a powerful modelling approach for understanding several aspects of spatial heterogeneity and nonlocality. In this paper, we propose an accurate, unconditionally stable, and maximum principle preserving fourth-order method both in space and time variables to study the complex dynamical processes of two-component nonlinear space-fractional reaction–diffusion systems posed in high dimensions. To achieve a fast fourth-order accurate method, we adapt the matrix transfer technique with a fast Fourier transform-based implementation in space and an exponential integrator in time. The main advantage of the method is that it avoids storing the large dense matrix resulting from discretizing the fractional operator with the matrix transform approach and significantly reduces the computational costs. Some numerical experiments are carried out in two and three space dimensions to demonstrate the accuracy and computational efficiency of the method.

KEYWORDS:

• Exponential integrator
• matrix transfer technique
• fast Fourier transform
• Gray–Scott model
• Schnakenberg model

YYYY AMS SUBJECT CLASSIFICATION:

• 65Mxx 2020

Acknowledgments

The author is grateful to the scholarly activities committee of Utah Valley University for providing the summer research award.

Disclosure statement

No potential conflict of interest was reported by the author(s).