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In this paper, a fast synthetic iterative scheme is developed to accelerateconvergence for the implicit DOM based on the stationary phonon BTE. The keyinnovative point of the present scheme is the introduction of the macroscopicsynthetic diffusion equation for the temperature, which is obtained from thezero- and first-order moment equations of the phonon BTE. The syntheticdiffusion equation, which is asymptomatically preserving to the Fourier's heatconduction equation in the diffusive regime, contains a term related to theFourier's law and a term determined by the second-order moment of thedistribution function that reflects the non-Fourier heat transfer. Themesoscopic kinetic equation and macroscopic diffusion equations are tightlycoupled together, because the diffusion equation provides the temperature forthe BTE, while the BTE provides the high-order moment to the diffusion equationto describe the non-Fourier heat transfer. This synthetic iterative schemestrengthens the coupling of all phonons in the phase space to facilitate thefast convergence from the diffusive to ballistic regimes. Typical numericaltests in one-, two-, and three-dimensional problems demonstrate that our schemecan 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 implicitDOM in the near-diffusive regime.

A fast synthetic iterative scheme for the stationary phonon Boltzmann transport equation