Implicit unified gas kinetic particle method for steady-state solution of multiscale phonon transport

This paper presents a highly efficient implicit unified gas-kinetic particle (IUGKP) method for obtaining steady-state solutions of multi-scale phonon transport. The method adapts and reinterprets the integral solution of the BGK equation for time-independent solutions. The distribution function at a given point is determined solely by the surrounding equilibrium states, where the corresponding macroscopic quantities are computed through a weighted sum of equilibrium distribution functions from neighboring spatial positions. From a particle perspective, changes in macroscopic quantities within a cell result from particle transport across cell interfaces. These particles are sampled according to the equilibrium state of their original cells, accounting for their mean free path as the traveling distance. The IUGKP method evolves the solution according to the physical relaxation time scale, achieving high efficiency in large Knudsen number regimes. To accelerate convergence for small Knudsen numbers, an inexact Newton iteration method is implemented, incorporating macroscopic equations for convergence acceleration in the near-diffusive limit. The method also addresses spatial-temporal inconsistency caused by relaxation time variations in physical space through the null-collision concept. Numerical tests demonstrate the method's excellent performance in accelerating multi-scale phonon transport solutions, achieving speedups of one to two orders of magnitude. The IUGKP method proves to be an efficient and accurate computational tool for simulating multiscale non-equilibrium heat transfer, offering significant advantages over traditional methods in both numerical performance and physical applicability.