Convergence of a Fast Explicit Operator Splitting Method for the Molecular Beam Epitaxy Model

TL;DR

This paper introduces a convergent, explicit operator splitting method for the molecular beam epitaxy model, combining spectral and finite difference schemes, with proven error bounds and numerical validation for stability and dynamics.

Contribution

It presents a new compact difference scheme for the nonlinear part and analyzes the convergence rate of the FEOS method, extending its applicability to large domains and complex dynamics.

Findings

The method achieves an error order of O(τ^2 + h^4) in L^2 norm.

Numerical experiments confirm the robustness for small coefficients.

Simulations reveal coarsening dynamics and 1/3 power laws in 2D.

Abstract

A fast explicit operator splitting (FEOS) method for the molecular beam epitaxy model has been presented in [Cheng, et al., Fast and stable explicit operator splitting methods for phase-field models, J. Comput. Phys., submitted]. The original problem is split into linear and nonlinear subproblems. For the linear part, the pseudo-spectral method is adopted; for the nonlinear part, a 33-point difference scheme is constructed. Here, we give a compact center-difference scheme involving fewer points for the nonlinear subproblem. Besides, we analyze the convergence rate of the algorithm. The global error order $\mathcal{O}(\tau^2+h^4)$ in discrete $L^2$-norm is proved theoretically and verified numerically. Some numerical experiments show the robustness of the algorithm for small coefficients of the fourth-order term for the one-dimensional case. Besides, coarsening dynamics are simulated in large domains and the $1/3$ power laws are observed for the two-dimensional case.