数学大讲堂

A general framework for high accuracy solutions to energy gradient flows from material science models

Abstract: A computational framework is presented for phase field materials science models that come from energy gradient flows. 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.

Fully implicit time stepping is used with adaptivity based on local error estimates. The implicit system at every time step is solved iteratively with Newton's iterations, with the linear systems solved at each step with a preconditioned conjugate gradient method. Solutions with high spatial and temporal accuracy are obtained. It is shown that large fully implicit time steps can be taken in times of meta-stable dynamics.

A comparison to time-stepping with operator splitting (into convex and concave parts that guarantees energy decrease in the numerical scheme) is done. It is shown numerically and analytically how these schemes can be inaccurate through meta-stable dynamics.