[Paper Review] A fast synthetic iterative scheme for the stationary phonon Boltzmann transport equation

Chuang Zhang, Songze Chen+2 more|arXiv (Cornell University)|Dec 16, 2018

Thermal properties of materialsMaterials Science53 references3 citations

TL;DR

This paper proposes a fast synthetic iterative scheme for solving the stationary phonon Boltzmann transport equation (BTE) using a tightly coupled macroscopic diffusion equation derived from zero- and first-order moment equations of the BTE. The method accelerates convergence by enhancing phonon coupling across phase space, achieving convergence in 100 or fewer iterations—1 to 3 orders of magnitude faster than conventional implicit discrete ordinate methods in the near-diffusive regime.

ABSTRACT

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.

Motivation & Objective

• Address the slow convergence of implicit discrete ordinate methods (DOM) in the near-diffusive regime for the stationary phonon Boltzmann transport equation (BTE).
• Overcome the inefficiency of traditional iterative schemes in coupling phonons across different wave vectors and spatial locations.
• Develop a synthetic scheme that unifies mesoscopic kinetic and macroscopic diffusion physics for improved convergence and accuracy.
• Enable efficient and accurate simulation of multiscale heat transfer from diffusive to ballistic regimes in 1D, 2D, and 3D problems.

Proposed method

• Derive a synthetic macroscopic diffusion equation from the zero- and first-order moment equations of the phonon BTE, ensuring asymptotic preservation of Fourier’s law in the diffusive limit.
• Decompose the heat flux in the diffusion equation into a Fourier term (dependent on temperature gradient) and a non-Fourier term (dependent on the second-order moment of the distribution function).
• Tightly couple the mesoscopic BTE and macroscopic diffusion equation: the BTE provides the second-order moment to the diffusion equation, while the diffusion equation supplies temperature to the BTE.
• Implement an iterative algorithm where the BTE is solved for each discrete wave vector, and the diffusion equation is updated using moment information, enabling fast global convergence.
• Use MPI parallelization across solid angle space to handle large-scale 3D simulations with high resolution.
• Ensure energy conservation and numerical stability by enforcing moment-based updates at each iteration step.

Experimental results

Research questions

• RQ1 Can a synthetic iterative scheme significantly accelerate convergence of the implicit DOM for the stationary phonon BTE in the near-diffusive regime?
• RQ2 How does coupling the mesoscopic BTE with a macroscopic diffusion equation derived from moment equations improve convergence speed and accuracy?
• RQ3 To what extent does the synthetic scheme maintain accuracy across the full range of transport regimes—from diffusive to ballistic—while reducing iteration counts?
• RQ4 Can the scheme handle large-scale 3D multiscale heat transfer problems efficiently with minimal computational overhead?
• RQ5 What is the quantitative improvement in convergence rate compared to conventional implicit DOM in realistic 3D configurations?

Key findings

• The synthetic iterative scheme achieves convergence in 72 to 99 iteration steps for all 3D test cases, including large-scale problems with up to 12×12×6 μm domains.
• Convergence is consistently reached within 100 iterations across all tested configurations, including 1D, 2D, and 3D problems with varying grid sizes and angular discretizations.
• The method accelerates convergence by 1 to 3 orders of magnitude compared to the conventional implicit DOM in the near-diffusive regime.
• The scheme accurately captures non-Fourier heat transfer effects, such as temperature jumps near hot sources, indicating failure of Fourier’s law in these regions.
• The synthetic diffusion equation correctly recovers the Fourier limit and accurately models both diffusive and non-diffusive heat transfer physics.
• The use of MPI parallelization enables efficient computation on 192 cores, making large-scale 3D simulations feasible with high resolution.