Proper generalized decomposition with synthetic preconditioning for multiscale parametric phonon transport

Abstract

Deterministic solution of the phonon Boltzmann transport equation is costly for two reasons: the distribution depends on high-dimensional phase space, and source iteration converges slowly when scattering is frequent. We address both for the steady linearized gray Callaway model in two dimensions by combining proper generalized decomposition (PGD) with the general synthetic iterative scheme (GSIS). The distribution is a sum of products of a space--azimuth factor and a polar factor. This splitting is intrinsically low-rank: along each characteristic the polar angle only stretches the in-plane optical depth, so the rank is bounded, uniformly in the Knudsen numbers, by the numerical rank of the kernel $\exp(-τ/\sinϕ)$. PGD--GSIS updates the factors against a macroscopic source frozen within each alternating step and refreshed by the GSIS moment equations. A Fourier analysis shows that pure PGD, although implicit in the polar factor, moves macroscopic information only through lagged spatial sweeps and stalls like source iteration, whereas the synthetic refresh sees only third-order moments of the representation error. On an isothermal cavity, cross-plane conduction and phonon Poiseuille flow, PGD--GSIS reproduces full-order solutions at ranks 3--14, with heat-flux errors below $6\times10^{-3}$ from the diffusive to the ballistic regime and memory independent of the polar resolution. Treating the inverse Knudsen numbers as coordinates of the polar factor yields one spatial basis over four decades of both; a joint update of all polar factors, which costs only small nodal linear systems, is required for accuracy at large Knudsen numbers.