High accuracy benchmark problems for Allen-Cahn and Cahn-Hilliard dynamics

There is a large literature of numerical methods for phase field models from materials science. The prototype models are the AllenCahn and Cahn-Hilliard equations. We present four benchmark problems for these equations, with numerical results validated using several computational methods with different spatial and temporal discretizations. Our goal is to provide the scientific community with a reliable reference point for assessing the accuracy and reliability of future software for this important class of problem.