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

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.