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