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

Energy Stable and Efficient Finite-Difference Nonlinear Multigrid Schemes for the Modified Phase Field Crystal Equation