  Ballistic phonon effects, arising on length scales comparable to the
mean-free-path, result in non-diffusive heat flow and alter the thermal
properties of materials. Simple theoretical models that accurately capture
non-diffusive transport physics are valuable for experimental analysis,
technology design, and providing physical insight. In this work, we utilize and
extend the McKelvey-Shockley (McK-S) flux method, a simple and accurate
framework, to investigate ballistic effects in transient phonon transport
submitted to a spatially sinusoidal heating profile, simulating a transient
thermal grating. We begin by extending a previous McK-S formulation to include
inelastic scattering, then obtain an analytical solution in the single phonon
energy case (gray approximation), and after show how this approach can readily
support a full phonon dispersion and mean-free-path distribution. The results
agree with experimental data and compare very well to solutions of the phonon
Boltzmann transport equation in the diffusive and weakly quasi-ballistic
transport regimes. We discuss the role of ballistic and non-equilibrium
physics, and show that inelastic scattering is key to retrieving the heat
equation solution in the diffusive limit. Overall the McK-S flux method, which
takes the form of a diffusion-like equation, proves to be a simple and accurate
framework that is applicable from the ballistic to diffusive transport regime.
