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We establish a general framework for developing, efficient energy stablenumerical schemes for gradient flows and develop three classes of generalizedscalar auxiliary variable approaches (G-SAV). Numerical schemes based on theG-SAV approaches are as efficient as the original SAV schemes\cite{SXY19,cheng2018multiple} for gradient flows, i.e., only require solvinglinear equations with constant coefficients at each time step, can beunconditionally energy stable. But G-SAV approaches remove the definitionrestriction that auxiliary variables can only be square root function. Thedefinition form of auxiliary variable is applicable to any reversible functionfor G-SAV approaches . Ample numerical results for phase field models arepresented to validate the effectiveness and accuracy of the proposed G-SAVnumerical schemes.

The generalized scalar auxiliary variable approach (G-SAV) for gradient flows