  We combine the theoretical method of calculating spin wave excitation with
the finite-temperature modeling and calculate the magnon-phonon relaxation time
in the technologically important material Yttrium iron garnet (YIG) from first
principles. The finite lifetime of magnon excitation is found to arise from the
fluctuation of the exchange interaction of magnetic atoms in YIG. At room
temperature, the magnon spectra have significant broadening that is used to
extract the magnon-phonon relaxation time quantitatively. The latter is a
phenomenological parameter of great importance in YIG-based spintronics
research. We find that the magnon-phonon relaxation time for the optical magnon
is a constant while that for the acoustic magnon is proportional to $1/k^2$ in
the long-wavelength regime.
