UGKWP and IUGKP methods for Multi-Scale Phonon Transport with Dispersion and Polarization

Abstract

This paper presents two novel methods for solving multi-scale phonon transport problems with dispersion and polarization effects: the unified gas-kinetic wave-particle (UGKWP) method and the implicit unified gas-kinetic particle (IUGKP) method. Both approaches are based on solving multiple groups of BGK equations at discrete frequency points. The UGKWP method constructs multiscale macroscopic fluxes at cell interfaces through the integral solution of the unsteady BGK equation and efficiently captures non-equilibrium transport using statistical particles. Its wave-particle adaptive framework ensures computational efficiency across different regimes: in the diffusive limit, it matches the cost of explicit diffusion equation solutions, while in the ballistic limit, it performs comparably to pure particle methods. The IUGKP method, specifically designed for steady-state problems, determines the particle evolution scale based on the physical mean free path. This approach enables rapid convergence at both large and small Knudsen numbers, with the latter facilitated by a newly constructed macroscopic prediction equation. Both methods incorporate an adaptive frequency-space sampling technique that maintains particle counts per cell comparable to single-frequency methods, significantly improving computational efficiency and memory usage. The accuracy and efficiency of both methods are validated through various numerical tests, including large-scale three-dimensional conduction heat transfer simulations. Results demonstrate their effectiveness in handling complex phonon transport phenomena across multiple scales.