  In this paper, we present a parallel numerical algorithm for solving the
phase field crystal equation. In the algorithm, a semi-implicit finite
difference scheme is derived based on the discrete variational derivative
method. Theoretical analysis is provided to show that the scheme is
unconditionally energy stable and can achieve second-order accuracy in both
space and time. An adaptive time step strategy is adopted such that the time
step size can be flexibly controlled based on the dynamical evolution of the
problem. At each time step, a nonlinear algebraic system is constructed from
the discretization of the phase field crystal equation and solved by a domain
decomposition based, parallel Newton--Krylov--Schwarz method with improved
boundary conditions for subdomain problems. Numerical experiments with several
two and three dimensional test cases show that the proposed algorithm is
second-order accurate in both space and time, energy stable with large time
steps, and highly scalable to over ten thousands processor cores on the Sunway
TaihuLight supercomputer.
