  Scalar auxiliary variable (SAV) methods are a class of linear schemes for
solving gradient flows that are known for the stability of a `modified' energy.
In this paper, we propose an improved SAV (iSAV) scheme that not only retains
the complete linearity but also ensures rigorously the stability of the
original energy. The convergence and optimal error bound are rigorously
established for the iSAV scheme and discussions are made for its high-order
extension. Extensive numerical experiments are done to validate the
convergence, robustness and energy stability of iSAV, and some comparisons are
made.
