ADAPTIVE SECOND-ORDER CRANK--NICOLSON TIME-STEPPING SCHEMES FOR TIME-FRACTIONAL MOLECULAR BEAM EPITAXIAL GROWTH MODELS.

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 fractional derivative. Based on the piecewise linear interpolation, the Caputo's 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 semidefinite property of the integral kernel. The resulting Crank--Nicolson SAV time-stepping schemes are unconditionally energy stable on arbitrary nonuniform time meshes. The fast algorithm and adaptive time strategy are employed to speed up the numerical computation. Ample numerical results show that our methods are computationally efficient in multiscale time simulations and appropriate for accurately resolving the intrinsically initial singularity of the solution and for efficiently capturing the fast dynamics away from the initial time. [ABSTRACT FROM AUTHOR]