  The phase-field crystal equation, a parabolic, sixth-order and nonlinear
partial differential equation, has generated considerable interest as a
possible solution to problems arising in molecular dynamics. This is because
the phase-field crystal model can capture atomic-scale effects at time-scales
that are orders of magnitude larger than what molecular dynamics simulations
can afford presently. Nonetheless, solving this equation is not a trivial task,
as a non-increasing free energy and mass conservation need to be verified for
the numerical solution to be valid. This work focuses on these issues, and
proposes a novel algorithm that guarantees mass conservation, unconditional
energy stability and is second-order accurate in time. This is achieved through
a convex-concave splitting of the nonlinearity present in the equation, along
with the use of a stabilization term that bounds possible increases in free
energy. We present numerical results that validate our mathematical proofs, and
show two and three dimensional simulations involving crystal growth that
showcase the robustness of the method.
