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Linear energy stable methods for an epitaxial growth model with slope selection

We consider a phase field model for molecular beam epitaxial growth with slope selection with the goal of determining linear energy stable time integration methods for the dynamics. Stable methods for this model have been found via a concave-convex splitting of the dynamics, but this approach generally leads to a nonlinear update equation. We seek a linear energy stable method to allow for simple and efficient time marching with fast Fourier transforms. Our approach is to parametrize a class of semi-implicit methods and perform unconditional von Neumann stability analysis to identify the region of stability in parameter space. Since unconditional von Neumann stability does not ensure energy stability, we perform extensive numerical tests and find strong agreement between the predicted and observed stable regions of parameter space. This analysis elucidates a novel feature that the stability region in parameter space differs for a mono-domain system (single equilibrium slope) versus a many-domain system (coarsening facets from an initially flat surface). The utility of these steps is then demonstrated by a comparison of the coarsening dynamics for isotropic and anisotropic variants of the model.