  The prediction of the melting curve of metals by extrapolation to high
pressures and temperatures based on the Lindemann-Gilvarry criterion (LG)
assuming harmonic Debye solid is presented. The LG formulation uses the bulk
modulus B and its pressure derivative B' as fit parameters deduced directly
from the equation of state, however, the results are not unique and strongly
depends on the chosen equation of state (EOS). By introducing a constraint that
the bulk moduli parameters B and B' must simultaneously obey the
Lindemann-Gilvarry criterion (LG) and the EOS, consistent bulk moduli are
derived. The cold pressure Pc and the cold melting curve are obtained by
introducing an effective Gr\"uneisen parameter ($\gamma$eff) to the LG
approximated equation together with the above constraint. It is claimed that
isochoric condition exists in diamond anvil cells (DAC), thus upon raising the
temperature and approaching the melt constant volume is maintained. Isochoric
condition in the DAC means that the developed thermal pressure (Pth) should be
accounted in the LG formulation. Therefor, the actual pressure (Pc+Pth) sensed
by the sample confined in the DAC should be inserted to the LG melting formula.
This brings along the demand that the shock waves Hugoniot melting data should
serve as anchor for deriving the correct melting curves of metals where the
Gr\"uneisen parameter $\gamma$ at ambient conditions ($\gamma$ o) is directly
determined. The melting curves up to ultra high pressures of Al, Cu, U and Fe
metals are presented and discussed. In this manner, special attention is given
to $\epsilon$-Fe as isobaric condition in the DAC has been claimed. Utilizing
the present approach we obtain the melting temperature of iron in the earth
inner core boundary (ICB, 330GPa) is 5900$\pm$100K.
