  An efficient implicit kinetic scheme is developed to solve the stationary
phonon Boltzmann transport equation (BTE) based on the non-gray model including
the phonon dispersion and polarization. Due to the wide range of the dispersed
phonon mean free paths, the phonon transport under the non-gray model is
essentially multiscale, and has to be solved differently and appropriately for
varied phonon frequencies and branches. The proposed implicit kinetic scheme is
composed of a microscopic iteration and a macroscopic iteration. The
microscopic iteration is capable of automatically adapting with varied phonon
mean free path of each phonon frequency and branch through solving the phonon
BTE. The energy transfer of all phonons is gathered together by the microscopic
iteration to evaluate the heat flux. The temperature field is predicted through
a macroscopic heat transfer equation according to the heat flux, and the
equilibrium state in the phonon BTE is also updated. The combination of the
phonon BTE solver and the macroscopic equation makes the present method very
efficient in a wide length scale. Three numerical tests, including the
cross-plane, in-plane and nano-porous heat transfer in silicon, validate that
the present scheme can handle with the phonon dispersion and polarization
correctly and predict the multiscale heat transfer phenomena efficiently in a
wide range. The present method could be tens of times faster than the typical
implicit DOM and keeps the same amount of the memory requirements as the
Fourier solver for multiscale heat transfer problem.
