An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems

Authors:
Ryan I. Fernandes;Graeme Fairweather
Affiliations:
Department of Mathematics, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates;Mathematical Reviews, American Mathematical Society, 416 Fourth Street, Ann Arbor, MI 48103, USA
Venue:
Journal of Computational Physics
Year:
2012

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Abstract

An alternating direction implicit (ADI) orthogonal spline collocation (OSC) method is described for the approximate solution of a class of nonlinear reaction-diffusion systems. Its efficacy is demonstrated on the solution of well-known examples of such systems, specifically the Brusselator, Gray-Scott, Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other numerical techniques considered in the literature. The new ADI method is based on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is efficient, requiring at each time level only O(N) operations where N is the number of unknowns. Moreover, it is shown to produce approximations which are of optimal global accuracy in various norms, and to possess superconvergence properties.