  In this paper we present two unconditionally energy stable finite difference
schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order
nonlinear damped wave equation, of which the purely parabolic Phase Field
Crystal (PFC) model can be viewed as a special case. The first is a convex
splitting scheme based on an appropriate decomposition of the discrete energy
and is first order accurate in time and second order accurate in space. The
second is a new, fully second-order scheme that also respects the convex
splitting of the energy. Both schemes are nonlinear but may be formulated from
the gradients of strictly convex, coercive functionals. Thus, both are uniquely
solvable regardless of the time and space step sizes. The schemes are solved by
efficient nonlinear multigrid methods. Numerical results are presented
demonstrating the accuracy, energy stability, efficiency, and practical utility
of the schemes. In particular, we show that our multigrid solvers enjoy
optimal, or nearly optimal complexity in the solution of the nonlinear schemes.
