Local Structure Preserving Algorithms for Molecular Beam Epitaxial Models with Slope Selection Based on Energy Quadratization Method

A molecular beam epitaxial (MBE) model with slope selection consisting of a fourth order Ginzburg-Landau double well potential, is derived from the variation of the free energy. One challenge in constructing the local structure preserving algorithms (SPAs) for the MBE model with slope selection is how to properly discretize the equation in space and time simultaneously in order to preserve the local structure at the discrete level. To resolve this issue, we employ the local energy dissipation property and the energy quadratization techniques. One novelty is that all nonlinear terms are treated semi-explicitly. The other novelty is that we introduce proper intermediate variables to make the space operators act on one single term, which is one crucial step in constructing local SPAs. We then develop two local energy dissipation preserving schemes and show rigorously the local energy dissipation property of the two schemes. Under suitable boundary conditions, such as periodic boundary conditions, the algorithms can preserve not only mass but also global energy dissipation property. Numerical experiments confirm the second-order accuracy and show the excellent performance of the schemes proposed.