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

Abstract

The heat transfer in solid materials at the micro- and nano-scale can be described by the mesoscopic phonon Boltzmann transport equation (BTE), rather than the macroscopic Fourier’s heat conduction equation that works only in the diffusive regime. The implicit discrete ordinate method (DOM) is efficient to find steady-state solutions to the BTE for highly non-equilibrium heat transfer problems, but converges extremely slowly in the near-diffusive regime. In this paper, a fast synthetic iterative scheme is developed to expedite convergence for the implicit DOM. The key innovative point of the present scheme is the introduction of a macroscopic diffusion-type equation for the temperature variation, which is exactly derived from the phonon BTE and valid for all Knudsen numbers. The synthetic equation, which is asymptomatically preserving to Fourier’s heat conduction equation in the diffusive regime, contains a term related to 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-type equations are tightly coupled, because the macroscopic equation provides the temperature for the BTE, while the BTE provides a high-order moment to the diffusion-type equation. This synthetic iterative scheme strengthens the coupling of phonons with different wave vectors in the phase space to facilitate 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, the present convergence is one to three orders of magnitude faster than the traditional implicit DOM in the near-diffusive regime.