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Adaptive second-order Crank-Nicolson time-stepping methods using the recentscalar auxiliary variable (SAV) approach are developed for the time-fractionalMolecular Beam Epitaxial models with Caputo's derivative. Based on thepiecewise linear interpolation, the Caputo's fractional derivative isapproximated by a novel second-order formula, which is naturally suitable for ageneral class of nonuniform meshes and essentially preserves the positivesemi-definite property of integral kernel. The resulting Crank-Nicolson SAVtime-stepping schemes are unconditional energy stable on nonuniform timemeshes, and are computationally efficient in multiscale time simulations whencombined with adaptive time steps, such as are appropriate for accuratelyresolving the intrinsically initial singularity of solution and for efficientlycapturing fast dynamics away from the initial time. Numerical examples arepresented to show the effectiveness of our methods.

Adaptive second-order Crank-Nicolson time-stepping schemes for time fractional molecular beam epitaxial growth models