  The direct simulation of the dynamics of second sound in graphitic materials
remains a challenging task due to lack of methodology for solving the phonon
Boltzmann equation in such a stiff hydrodynamic regime. In this work, we aim to
tackle this challenge by developing a multiscale numerical scheme for the
transient phonon Boltzmann equation under Callaway's dual relaxation model
which captures well the collective phonon kinetics. Comparing to traditional
numerical methods, the present multiscale scheme is efficient, accurate and
stable in all transport regimes attributed to avoiding the use of time and
spatial steps smaller than the relaxation time and mean free path of phonons.
The formation, propagation and composition of ballistic pulses and second sound
in graphene ribbon in two classical paradigms for experimental detection are
investigated via the multiscale scheme. The second sound is declared to be
mainly contributed by ZA phonon modes, whereas the ballistic pulses are mainly
contributed by LA and TA phonon modes. The influence of temperature, isotope
abundance and ribbon size on the second sound propagation is also explored. The
speed of second sound in the observation window is found to be at most 20
percentages smaller than the theoretical value in hydrodynamic limit due to the
finite umklapp, isotope and edge resistive scattering. The present study will
contribute to not only the solution methodology of phonon Boltzmann equation,
but also the physics of transient hydrodynamic phonon transport as guidance for
future experimental detection.
