  The scalar auxiliary variable (SAV) approach is a very popular and efficient
method to simulate various phase field models. To save the computational cost,
a new SAV approach is given by introducing a new variable $\theta$. The new SAV
approach can be proved to save nearly half CPU time of the original SAV
approach while keeping all its other advantages. In this paper, we propose a
novel technique to construct an exponential semi-implicit scalar auxiliary
variable (ESI-SAV) approach without introducing any extra variables. The new
proposed method also only needs to solve one linear equation with constant
coefficients at each time step. Furthermore, the constructed ESI-SAV method
does not need the bounded below restriction of nonlinear free energy potential
which is more reasonable and effective for various phase field models.
Meanwhile it is easy to construct first-order, second-order and higher-order
unconditionally energy stable time-stepping schemes. Other than that, the
ESI-SAV approach can be proved to be effective to solve the non-gradient but
dissipative system such as Navier-Stokes equations. Several numerical examples
are provided to demonstrate the improved efficiency and accuracy of the
proposed method.
