Energy dissipation–preserving time-dependent auxiliary variable method for the phase-field crystal and the Swift–Hohenberg models

In this study, we develop first- and second-order time-accurate energy stable methods for the phase-field crystal equation and the Swift–Hohenberg equation with quadratic-cubic non-linearity. Based on a new Lagrange multiplier approach, the first-order time-accurate schemes dissipate the original energy in a time-discretized version, which are different from the modified energy laws obtained by the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) methods. Moreover, the proposed schemes do not require the bounded-from-below restriction which is necessary in the IEQ or SAV approach. We rigorously prove the energy dissipations of first- and second-order accurate methods with respect to the original energy and pseudo-energy in the time-discretized versions, respectively. An efficient algorithm is used to decouple the resulting weakly coupled systems. In one time iteration, only two linear systems with constant coefficients and one non-linear algebraic equation need to be solved. Finally, the accuracy, stability and practicability of the proposed methods are validated by intensive numerical tests.