  In this paper, we present a novel investigation of the so-called SAV
approach, which is a framework to construct linearly implicit geometric
numerical integrators for partial differential equations with variational
structure. SAV approach was originally proposed for the gradient flows that
have lower-bounded nonlinear potentials such as the Allen-Cahn and
Cahn-Hilliard equations, and this assumption on the energy was essential. In
this paper, we propose a novel approach to address gradient flows with
unbounded energy such as the KdV equation by a decomposition of energy
functionals. Further, we will show that the equation of the SAV approach, which
is a system of equations with scalar auxiliary variables, is expressed as
another gradient system that inherits the variational structure of the original
system. This expression allows us to construct novel higher-order integrators
by a certain class of Runge-Kutta methods. We will propose second and fourth
order schemes for conservative systems in our framework and present several
numerical examples.
