Error analysis of positivity-preserving energy stable schemes for the modified phase field crystal model

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In this paper, we introduce second-order numerical schemes for the modified phase field crystal (MPFC) model that are decoupled, linear, positivity-preserving, and unconditionally energy-stable. These schemes adopt a positivity-preserving auxiliary variable method to explicitly handle the nonlinear potential function, resulting in decoupled linear systems with constant coefficients at each time step. We rigorously demonstrate that the auxiliary variables remain positive throughout all time steps and prove the unconditionally energy stability of these schemes. The stability pertains to a discrete modified energy, rather than the original free energy or the pseudo energy of the MPFC system. Moreover, a detailed error analysis is provided. A series of numerical experiments are conducted to validate the accuracy and efficiency of our proposed schemes.