High accuracy solutions to energy gradient flows from material science models

A computational framework is presented for materials science models that come from energy gradient flows. The models of interest lead to the evolution of structure involving two or more phases. The framework includes higher order derivative models and vector problems. Solutions are considered in periodic cells and standard Fourier spectral discretization in space is used. Implicit time stepping is used with adaptivity based on local error estimates. The implicit system at every time step is solved iteratively with Newtonʼs method. The resulting linear systems are solved in inner iterations with the conjugate gradient method, using a novel preconditioner that is a constant coefficient version of the system, taking values for the coefficients at the pure phase states. Solutions with high spatial and temporal accuracy are obtained. The dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model (that represents the scaled width of transition layers between phases) is investigated numerically and with formal asymptotics in a simple setting. The asymptotic results require a conjecture on the rank of a modified square distance matrix. Results from a fast, graphical processing unit implementation for a three-dimensional model are shown. A comparison to time stepping with operator splitting (into convex and concave parts that guarantees energy decrease in the numerical scheme) is done.