  In this paper, a fast synthetic iterative scheme is developed to accelerate
convergence for the implicit DOM based on the stationary phonon BTE. The key
innovative point of the present scheme is the introduction of the macroscopic
synthetic diffusion equation for the temperature, which is obtained from the
zero- and first-order moment equations of the phonon BTE. The synthetic
diffusion equation, which is asymptomatically preserving to the Fourier's heat
conduction equation in the diffusive regime, contains a term related to the
Fourier's law and a term determined by the second-order moment of the
distribution function that reflects the non-Fourier heat transfer. The
mesoscopic kinetic equation and macroscopic diffusion equations are tightly
coupled together, because the diffusion equation provides the temperature for
the BTE, while the BTE provides the high-order moment to the diffusion equation
to describe the non-Fourier heat transfer. This synthetic iterative scheme
strengthens the coupling of all phonons in the phase space to facilitate the
fast convergence from the diffusive to ballistic regimes. Typical numerical
tests in one-, two-, and three-dimensional problems demonstrate that our scheme
can describe the multiscale heat transfer problems accurately and efficiently.
For all test cases convergence is reached within one hundred iteration steps,
which is one to three orders of magnitude faster than the traditional implicit
DOM in the near-diffusive regime.
