标题：Fractional Gray-Scott model: Well-posedness, discretization, and simulations
作者：Wang, Tingting; Song, Fangying; Wang, Hong; Karniadakis, George Em
作者机构：[Wang, Tingting] Shandong Univ, Sch Math, Jinan 250100, Shandong, Peoples R China.; [Wang, Tingting; Karniadakis, George Em] Brown Univ, Div Appl Ma 更多
通讯作者：Karniadakis, George Em;Karniadakis, GE
通讯作者地址：[Karniadakis, GE]Brown Univ, Div Appl Math, Providence, RI 02912 USA.
来源：COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
关键词：Pattern formation; ADI algorithm; Anomalous transport; Finite; difference; Spectral collocation; Radial distribution function
摘要：The Gray-Scott (GS) model represents the dynamics and steady state pattern formation in reaction-diffusion systems and has been extensively studied in the past. In this paper, we consider the effects of anomalous diffusion on pattern formation by introducing the fractional Laplacian into the GS model. First, we prove the well-posedness of the fractional GS model. We then introduce the Crank-Nicolson (C-N) scheme for time discretization and weighted shifted Grunwald difference operator for spatial discretization. We perform stability analysis for the time semi-discrete numerical scheme, and furthermore, we analyze numerically the errors with benchmark solutions that show second-order convergence both in time and space. We also employ the spectral collocation method in space and C-N scheme in time to solve the GS model in order to verify the accuracy of our numerical solutions. We observe the formation of different patterns at different values of the fractional order, which are quite different from the patterns of the corresponding integer-order GS model, and quantify them by using the radial distribution function (RDF). Finally, we discover the scaling law for steady patterns of the RDFs in terms of the fractional order. (C) 2019 Elsevier B.V. All rights reserved.