The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank-Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h ²) and linearized Crank-Nicolson scheme is convergent with the convergence order of O(τ ² + h ²) in discrete L ²-norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis.