Fractional Gray–Scott model: Well-posedness, discretization, and simulations

• Published in: Computer methods in applied mechanics and engineering Vol. 347; pp. 1030 - 1049
• Main Authors: Wang, Tingting, Song, Fangying, Wang, Hong, Karniadakis, George Em
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.04.2019; Elsevier BV
• Subjects: ADI algorithm; Anomalous transport; Collocation methods; Computer simulation; Diffusion effects; Discretization; Distribution functions; Finite difference; Finite differences; Mathematical models; Model accuracy; Operators (mathematics); Pattern formation; Radial distribution; Radial distribution function; Scaling laws; Spectral collocation; Stability analysis; Well posed problems; Finite difference; Pattern formation; Spectral collocation; Anomalous transport; Radial distribution function; ADI algorithm
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2019.01.002

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 Grünwald 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. •We consider the effects of anomalous diffusion on pattern formation by fractional GS model.•We prove the well-posedness of the fractional GS model.•A numerical algorithm is studied to solve the fractional GS model.•We present numerical simulations of the fractional GS model, and a scaling law for steady patterns of RDFs.