A Coupled Ordinates Method for Convergence Acceleration of the Phonon Boltzmann Transport Equation

Abstract

Sequential solution methods are commonly-used for solving the phonon Boltzmann transport equation (BTE) because of simplicity of implementation and low storage requirements. However, they exhibit poor convergence for low Knudsen numbers. This is because sequential solution procedures couple the phonon BTEs in physical space efficiently but the coupling is inefficient in wave-vector (K) space. As the Knudsen number decreases, coupling in K space becomes dominant and convergence rates fall. Since materials like silicon have K-resolved Knudsen numbers that span 3–4 orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, non-gray solutions of the BTE almost always experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for solving the phonon BTE in the relaxation time approximation. Here, inter-equation coupling is treated implicitly through a point-coupled direct solution of the K-resolved BTEs at each control volume. This implicit solution is used as a relaxation sweep in a geometric multigrid method. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration, between 10 to 300 times, over the sequential procedure is found for heat conduction problems.