Efficient Matching Boundary Conditions of Two-dimensional Honeycomb Lattice for Atomic Simulations

TL;DR

This paper introduces a new set of simple, efficient boundary conditions for 2D honeycomb lattice simulations that effectively reduce boundary reflections across various wave types, improving computational accuracy.

Contribution

It develops explicit, high-efficiency matching boundary conditions for 2D honeycomb lattices, enhancing suppression of boundary reflections in atomic simulations.

Findings

High-order boundary conditions suppress both short and long waves effectively.

Boundary conditions are stable and maintain low kinetic energy decay.

Numerical tests confirm improved simulation accuracy.

Abstract

In this paper, we design a series of matching boundary conditions for a two-dimensional compound honeycomb lattice, which has an explicit and simple form, high computing efficiency and good effectiveness of suppressing boundary reflections. First, we formulate the dynamic equations and calculate the dispersion relation for the harmonic honeycomb lattice, then symmetrically choose specific atoms near the boundary to design different forms of matching boundary conditions. The boundary coefficients are determined by matching a residual function at some selected wavenumbers. Several atomic simulations are performed to test the effectiveness of matching boundary conditions in the example of a harmonic honeycomb lattice and a nonlinear honeycomb lattice with the FPU-$\beta$ potential. Numerical results illustrate that low-order matching boundary conditions mainly treat long waves, while the high-order matching boundary conditions can efficiently suppress short waves and long waves simultaneously. Decaying kinetic energy curves indicate the stability of matching boundary conditions in numerical simulations.