标题：A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations
作者：Zhang H.; Jiang X.; Zeng F.; Karniadakis G.E.
作者机构：[Zhang, H] School of Mathematics, Shandong University, Jinan, 250100, China;[ Jiang, X] School of Mathematics, Shandong University, Jinan, 250100, Chi 更多
通讯作者地址：[Jiang, X] School of Mathematics, Shandong UniversityChina;
来源：Journal of Computational Physics
关键词：Fourier spectral method; Linear stability; Optimal error estimate; Semi-implicit time-stepping method; Space-fractional reaction-diffusion equations
摘要：The reaction-diffusion model can generate a wide variety of spatial patterns, which has been widely applied in chemistry, biology, and physics, even used to explain self-regulated pattern formation in the developing animal embryo. In this work, a second-order stabilized semi-implicit time-stepping Fourier spectral method for the reaction-diffusion systems of equations with space described by the fractional Laplacian is developed. We adopt the temporal-spatial error splitting argument to illustrate that the proposed method is stable without imposing the CFL condition, and an optimal L2-error estimate in space is proved. We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical criterion to choose the time step size to guarantee the stability of the semi-implicit method. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen-Cahn, Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator, which are quite different from the patterns of the corresponding integer-order model. © 2019 Elsevier Inc.