  Adaptive second-order Crank-Nicolson time-stepping methods using the recent
scalar auxiliary variable (SAV) approach are developed for the time-fractional
Molecular Beam Epitaxial models with Caputo's derivative. Based on the
piecewise linear interpolation, the Caputo's fractional derivative is
approximated by a novel second-order formula, which is naturally suitable for a
general class of nonuniform meshes and essentially preserves the positive
semi-definite property of integral kernel. The resulting Crank-Nicolson SAV
time-stepping schemes are unconditional energy stable on nonuniform time
meshes, and are computationally efficient in multiscale time simulations when
combined with adaptive time steps, such as are appropriate for accurately
resolving the intrinsically initial singularity of solution and for efficiently
capturing fast dynamics away from the initial time. Numerical examples are
presented to show the effectiveness of our methods.
