| Summary: | 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 Cahn-Hillard equation. The transition layers between phases evolve over time and can move very slowly. The models have an order parameter epsilon. Fully developed transition layers have a width that scales linearly with epsilon. As epsilon goes to 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 dependence on epsilon and time step size is identified. The Allen-Cahn and Cahn-Hilliard equations have been used extensively to model phenomena in materials science. We strongly believe that our high order adaptive approach is also easily extensible to higher order models with application to pore formation in functionalized polymers and to cancerous tumor growth simulation. This is the subject of ongoing research.
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