  In this paper, we did a systematic comparative study on the accuracy of two
computational methods of elastic constants combined with the density functional
theory (DFT), the stress-strain method and the energy-strain method. We took
metal Ru as a prototype to compare its high-pressure elastic constants
calculated by our present stress-strain method with the previous energy-strain
results by others. Although the two methods yielded almost the same accuracy of
high-pressure elastic constants for Ru, our stress-strain method directly based
on the Hooke's law of elasticity theory is much straightforward and simple to
implement. However, the energy-strain method needs complicated pressure
corrections because of the pressure effects on the total energy. Various
crystal systems have various pressure correction methods. Hence, the
stress-strain method is preferred to calculate the high-pressure elastic
constants of materials. Furthermore, we analyzed the variations of the elastic
moduli, elastic anisotropy, sound velocities, Debye temperature of Ru with
pressure.
