Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces

TL;DR

This paper rigorously derives a continuum PDE model for step motion on vicinal surfaces from a mesoscopic BCF model, proving convergence as the lattice parameter approaches zero with a first-order rate.

Contribution

It provides a rigorous mathematical proof of the continuum limit of a mesoscopic step motion model, establishing convergence to the PDE solution.

Findings

Convergence of the discrete model to the PDE as lattice parameter approaches zero.

First-order convergence rate established for the continuum limit.

Validation of the PDE as an accurate continuum approximation for the mesoscopic model.

Abstract

This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.