We investigate an instability that leads to the filamentation of large-amplitude Alfvén waves propagating along a homogeneous magnetic field in weakly ionized plasmas by using a neutral MHD simulation code. The dispersion relation of small-amplitude Alfvén waves is confirmed by a one-dimensional simulation code. The linear instability leading to the filamentation is investigated by imposing small density perturbations across a magnetic field. We demonstrate the results of the nonlinear stage of the above filamentation instability with a two-dimensional simulation showing the occurrence of localized plasma heating during repeated coalescence and fission of the filaments. We discuss the implication of these results for plasma-heating processes observed in the photosphere of the Sun.