High-order time-adaptive numerical methods for the Allen-Cahn and Cahn-Hilliard equations

In some nonlinear reaction-diffusion equations of interest in applications, there are transition layers in solutions that separate two or more materials or phases in a medium when the reaction term is very large. Two well known equations that are of this type: The Allen-Cahn equation and the CahnHillard equation. The transition layers between phases evolve over time and can move very slowly. The models have an order parameter . Fully developed transition layers have a width that scales linearly with . As → 0, the time scale of evolution can also change and the problem becomes numerically challenging. We consider several numerical methods to obtain solutions to these equations, in order to build a robust, efficient and accurate numerical strategy. Explicit time stepping methods have severe time step constraints, so we direct our attention to implicit schemes. Second and third order time-adaptive methods are presented using spectral discretization in space. The implicit problem is solved using the conjugate gradient method with a novel preconditioner. The behaviour of the preconditioner is investigated, and the