  The charge-density-wave (CDW) order with macroscopically occupied electrons
distorts the underlying lattice and usually causes the softening of the
associated phonon mode. However, previous studies demonstrated that the
spin-Peierls transition does not always induce an associated phonon softening,
but the central-peak scenario applied in the quasi-one-dimensional compound
CuGeO$_3$. We generalize the lattice-dynamics studies on the two-dimensional
CDW state at van-Hove-singularity (VHS) filling and find that the CDW ordering
could develop a central peak at zero frequency while the associated phonon
undergoes hardening. The particle-hole scatterings between VHS points give rise
to a low-energy increased charge-density susceptibility, and their coupling to
the lattice dynamics induces two poles in the Green function for the
CDW-associated phonon mode. The zero-frequency pole corresponds to the
collective charge-density and phonon coupling mode. The high-frequency one is
related to the high-temperature phonon mode that hardens as reaching the CDW
transition. Our result may have the potential implication for the recently
discovered Kagome metal $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs) in which no soft
phonon is observed during the CDW transition.
