Stable Numerical Integration of an Epitaxial Growth Model with Slope Selection

TL;DR

This paper develops and analyzes stable, efficient semi-implicit numerical methods for simulating crystal growth in epitaxial models, ensuring stability through theoretical analysis and numerical validation.

Contribution

It introduces a class of semi-implicit methods linear in the updated field, enabling efficient Fourier-based implementation and stability analysis for epitaxial growth models.

Findings

Unconditional von Neumann stability regions identified.

Strong agreement between theoretical stability analysis and numerical tests.

Efficient simulation of crystal growth dynamics achieved.

Abstract

We consider a continuum phase field model for crystal growth via molecular beam epitaxy, with the goal of determining stable numerical time integration methods for the dynamics. We parametrize a class of semi-implicit methods that are linear in the updated field, which allows for efficient implementation with fast Fourier transforms. We perform unconditional von Neumann stability analysis to identify the region of stability in parameter space, and then test these predictions numerically for gradient stability. We find strong agreement between the approaches.