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

We consider the approximate solution of nonlinear reaction-diffusion systems on evolving domains that arise in a variety of areas including biology, chemistry, ecology and physics. By mapping a fixed domain onto the evolving domain at each time level, we generalize to evolving domains the ADI extrapolated Crank-Nicolson orthogonal spline collocation technique developed in [8], [9] for fixed domains. The new method is tested on the Schnakenberg model and we demonstrate numerically that it preserves the second-order accuracy in time and optimal accuracy in space for piecewise Hermite cubics in various norms. Moreover, the efficacy of the method is demonstrated on several test problems from the literature which involve various types of domain evolution but for which exact solutions are not known.