Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection

Abstract

Based on the local energy dissipation property of the molecular beam epitaxy (MBE) model with slope selection, we develop three, second order fully discrete, local energy dissipation rate preserving (LEDP) algorithms for the model using finite difference methods. For periodic boundary conditions, we show that these algorithms are global energy dissipation rate preserving (GEDP). For adiabatic, physical boundary conditions, we construct two GEDP algorithms from the three LEDP ones with consistently discretized physical boundary conditions. In addition, we show that all the algorithms preserve the total mass at the discrete level as well. Mesh refinement tests are conducted to confirm the convergence rates of the algorithms and two benchmark examples are presented to show the accuracy and performance of the methods.

A Hessian Recovery Based Linear Finite Element Method for Molecular Beam Epitaxy Growth Model with Slope Selection

“…There are also some other energy decay preserving numerical algorithms, for example, Chen et al in [8] presented a new multi-step Crank-Nicolson, which is proved to possess properties of total mass conservation and unconditionally energy stability. Lu et al in [27] developed a second-and third-order fully discrete scheme for (1.1) based on the local structure-preserving algorithm. Cheng and Wang in [7] gave a error estimate of second order accurate SAV numerical method for the epitaxial thin film equation.…”

“…In [32], Qiao et al established the error estimation of the mixed element method for (1.1). For the MBE model (1.1) with the Neumann boundary condition, Xia [45] developed a fully discrete stable discontinuous Galerkin method, and Lu et al in [27] proposed a finite difference scheme.…”

Local Energy Dissipation Rate Preserving Approximations to Driven Gradient Flows with Applications to Graphene Growth

“…How to design a local structure-preserving schemes that are also global structure-preserving for a given physical boundary conditions however remains an open problem though. In [64], we presented a few cases where the global energy-dissipation-rate schemes can be constructed using a combination of local energy-dissipation-rate preserving schemes, where different LEDRP schemes are adopted at different part of boundaries.…”

“…Finally, numerical simulations on growth of graphene sheets on copper are carried out and benchmarked against existing results in the literature. We note that this numerical paradigm can be applied to any partial differential equation system with deduced equations, not limited to thermodynamically consistent gradient flow models [64].…”