  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.
