A second-order linear energy-stable scheme for slope-selection epitaxial thin-film growth

The slope-selection epitaxial thin-film growth model was formulated as the $L^2$-gradient flow of an energy functional with a nonlinear potential of the surface slope. We developed a second-order, linear, and energy-stable scheme for this model based on a linear convex splitting in which the nonlinear potential term was treated explicitly together with an auxiliary term that ensured the convexity of the explicit part. The scheme was constructed using a second-order strong-stability-preserving implicit–explicit Runge–Kutta method. We explicitly identified the range of Runge–Kutta coefficients for which the original discrete energy decay property held, and proved that the scheme was unconditionally energy-stable with respect to the original discrete energy functional. Numerical results were presented to verify the accuracy, computational efficiency, and long-time energy stability of the scheme.