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

## TL;DR

This paper introduces adaptive second-order Crank-Nicolson schemes using the SAV approach for efficient and stable simulation of time-fractional molecular beam epitaxial growth models, especially on nonuniform meshes.

## Contribution

It develops a novel second-order approximation for Caputo's fractional derivative compatible with nonuniform meshes and demonstrates unconditional energy stability of the schemes.

## Key findings

- Schemes are unconditionally energy stable.
- Effective in multiscale time simulations.
- Accurately captures initial singularity and fast dynamics.

## Abstract

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.