  Nonlocal thermal transport is generally described by the Peierls-Boltzmann
transport equation (PBE). However, solving the PBE for a general space-time
dependent problem remains a challenging task due to the high dimensionality of
the integro-differential equation. In this work, we present a direct solution
to the space-time dependent PBE with a linearized collision matrix using an
eigendecomposition method. We show that there exists a generalized Fourier type
relation that links heat flux to the local temperature, and this constitutive
relation defines a thermal conductivity that depends on both time and space.
Combining this approach with ab initio calculations of phonon properties, we
demonstrate that the space-time dependent thermal conductivity gives rise to an
oscillatory response in temperature in a transient grating geometry in high
thermal conductivity materials. The present solution method allows us to extend
the reach of our computational capability for heat conduction to space-time
dependent nondiffusive transport regimes. This capability will not only enable
a more accurate interpretation of thermal measurements that observe nonlocal
thermal transport but also enhance our physical understanding of nonlocal
thermal transport in high thermal conductivity materials that are promising
candidates for nanoscale thermal management applications.
