Since their introduction over fifty years ago, alternating direction implicit (ADI) methods have been employed to solve a variety of time-dependent multidimensional problems. Their primary attraction is that they reduce such a problem to independent systems of one-dimensional problems.
In this paper, we describe an ADI method for the solution of a class of two-component nonlinear reaction-diffusion problems on evolving domains. In this method, orthogonal spline collocation (OSC) is used for the spatial discretization, and the time-stepping is done with an algebraically linear ADI method based on an extrapolated Crank-Nicolson OSC method. Numerical results are provided to demonstrate the expected global rates of convergence of the method.