  This study investigates the atomistic spin system in $\rm CrCl_{3}$, which
exhibits topologically nontrivial meron structures within its layered hexagonal
lattice framework. We analyze the complete model of discrete spin dynamics on a
two-dimensional hexagonal lattice and demonstrate its convergence to the
continuum Landau-Lifshitz-Gilbert equation in the weak sense. The primary
challenge lies in defining appropriate difference quotient and interpolation
operators for the hexagonal lattice since the loss of symmetry. To address
these, we utilized a one-step difference quotient for the 2nd nearest neighbors
and introduced novel multi-step difference quotients for the 1st and 3rd
nearest neighbors, enabling the integration by parts formula. Additionally, we
generalized Ladysenskaya's interpolation operator for hexagonal lattices and
provided an alternative strategy for the convergence procedure by applying an
isometric mapping property. This work provides necessary tools for analyzing
weak convergence in other atomistic nonlinear problems on hexagonal lattices
towards the continuum limit.
