  In this paper, we propose a novel family of high-order numerical schemes for
the gradient flow models based on the scalar auxiliary variable (SAV) approach,
which is named the high-order scalar auxiliary variable (HSAV) method. The
newly proposed schemes could be shown to reach arbitrarily high order in time
while preserving the energy dissipation law without any restriction on the time
step size (i.e., unconditionally energy stable). The HSAV strategy is rather
general that it does not depend on the specific expression of the effective
free energy, such that it applies to a class of thermodynamically consistent
gradient flow models arriving at semi-discrete high-order energy-stable
schemes. We then employ the Fourier pseudospectral method for spatial
discretization. The fully discrete schemes are also shown to be unconditionally
energy stable. Furthermore, we present several numerical experiments on several
widely-used gradient flow models, to demonstrate the accuracy, efficiency and
unconditionally energy stability of the HSAV schemes. The numerical results
verify that the HSAV schemes can reach the expected order of accuracy, and it
allows a much larger time step size to reach the same accuracy than the
standard SAV schemes.
