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In this paper, we present a parallel numerical algorithm for solving thephase field crystal equation. In the algorithm, a semi-implicit finitedifference scheme is derived based on the discrete variational derivativemethod. Theoretical analysis is provided to show that the scheme isunconditionally energy stable and can achieve second-order accuracy in bothspace and time. An adaptive time step strategy is adopted such that the timestep size can be flexibly controlled based on the dynamical evolution of theproblem. At each time step, a nonlinear algebraic system is constructed fromthe discretization of the phase field crystal equation and solved by a domaindecomposition based, parallel Newton--Krylov--Schwarz method with improvedboundary conditions for subdomain problems. Numerical experiments with severaltwo and three dimensional test cases show that the proposed algorithm issecond-order accurate in both space and time, energy stable with large timesteps, and highly scalable to over ten thousands processor cores on the SunwayTaihuLight supercomputer.

Parallel energy-stable phase field crystal simulations based on domain decomposition methods