A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect

In this paper, we extend the phase field crystal model to the modified phase field crystal model which includes diffusive dynamics, elastic interaction, and stochastic noises effect. We present a second-order accurate semi-implicit finite difference scheme for the modified phase field crystal model. The resulting scheme is based on the stabilized splitting method and Crank–Nicolson method. The nonlinear term is linearized by the Taylor series. The resulting scheme is linear at each time step, which makes it easy to be implemented and efficient to be solved by using the linear multigrid solver. We prove that the resulting scheme is unconditionally energy stable. Various numerical experiments are conducted to verify the accuracy and efficiency of our proposed algorithm.