Special Seminar in Applied Mathematics
A computational framework is presented for materials science models that come from energy gradient flows that lead to the evolution of structure involving two or more phases. The models are considered in periodic cells and standard Fourier spectral discretization in space is used. Implicit time stepping is used, and the resulting implicit systems are solved iteratively with a preconditioned conjugate gradient method. 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 framework is easily extended to higher order derivative models, higher dimensional settings, and vector problems. Several examples of its application are demonstrated, including a sixth order problem and a vector fourth order problem. A GPU implementation is described briefly. A comparison to time-stepping with operator splitting (into convex and concave parts) is done. It is shown that operator splitting, which is commonly used in practice and has guaranteed energy decrease per time step, severely limits the accuracy in meta-stable evolution.