Multirate integration of axisymmetric step-flow equations

TL;DR

This paper introduces a multirate integration method tailored for axisymmetric step-flow equations in surface evolution, effectively handling singularities and local stiffness to improve accuracy and efficiency.

Contribution

A novel multirate method designed for axisymmetric step-flow ODEs that manages singularities and stiffness while maintaining fourth order accuracy.

Findings

Successfully handles singular step trajectories.

Effectively manages local stiffness during step bunching.

Achieves fourth order accuracy in numerical solutions.

Abstract

We present a multirate method that is particularly suited for integrating the systems of Ordinary Differential Equations (ODEs) that arise in step models of surface evolution. The surface of a crystal lattice, that is slightly miscut from a plane of symmetry, consists of a series of terraces separated by steps. Under the assumption of axisymmetry, the step radii satisfy a system of ODEs that reflects the steps' response to step line tension and step-step interactions. Two main problems arise in the numerical solution of these equations. First, the trajectory of the innermost step can become singular, resulting in a divergent step velocity. Second, when a step bunching instability arises, the motion of steps within a bunch becomes very strongly stable, resulting in "local stiffness". The multirate method introduced in this paper ensures that small time steps are taken for singular and locally stiff components, while larger time steps are taken for the remaining ones. Special consideration is given to the construction of high order interpolants during run time which ensures fourth order accuracy of scheme for components of the solution sufficiently far away from singular trajectories.