General synthetic iterative scheme for fast solving coupled electron–phonon Boltzmann equations

Abstract

Understanding and controlling electron–phonon interactions is essential for optimizing thermal performance in microelectronic and thermoelectric devices, where strong non-equilibrium effects span multiple length and time scales. The coupled electron–phonon Boltzmann equations offer an insightful framework for modeling such transport, but their high dimensionality and stiffness pose significant computational challenges. Conventional iterative solvers typically suffer from excessive numerical dissipation and slow convergence in diffusive regimes. In this work, we develop a general synthetic iterative scheme (GSIS) that significantly accelerates the solution of coupled electron–phonon Boltzmann equations. The key innovation is the formulation of macroscopic synthetic equations that effectively capture diffusion-limit behavior, while precisely incorporating non-equilibrium corrections extracted from the kinetic level. During iterations, the kinetic solver provides high-order moment closures, while the macroscopic equations update the driving fields, ensuring rapid convergence. This strategic coupling across scales facilitates efficient global information exchange and robust multiscale resolution. Fourier analysis and numerical benchmarks show that GSIS can reduce computational time by up to three orders of magnitude compared to the conventional iterative scheme. Implemented with a high-order discontinuous Galerkin method, the GSIS is applicable to complex geometries and extendable to nonlinear and non-gray transport models.