An accurate and efficient algorithm for the time-fractional molecular beam epitaxy model with slope selection

Abstract

In this paper, we propose a time-fractional molecular beam epitaxy (MBE) model with slope selection using the classical Caputo fractional derivative of order $\alpha$ ($0<\alpha <1$), which is shown to possess an energy dissipation law. Then we develop its efficient and accurate, fully discrete, linear numerical approximation. Utilizing the classical $L^1$ numerical treatment for the time-fractional derivative and the invariant energy quadratization strategy, the resulted semi-discrete scheme is shown to preserve the energy dissipation law and the total mass in the time discrete level. The rigorous numerical analysis is also provided to show its first order convergence in time. The semi-discrete scheme is further discretized in space using the Fourier spectral method, resulting in a fully discrete linear scheme. The fast algorithm for approximating the time-fractional derivative is also introduced to result in an efficient full discrete linear scheme. Time refinement tests are conducted for both schemes, verifying their first order of time convergence for arbitrary fractional order $\alpha \in (0,1]$. Several numerical simulations are presented to demonstrate the accuracy and efficiency of the newly proposed schemes. By exploring the fast algorithm calculating the Caputo fractional derivative, our numerical scheme makes it practical for long time simulation of the MBE model while preserving its energy stability, which is essential for MBE model predictions. With the proposed fractional MBE model, we observe that the effective energy decaying scales as $O(t^{-}\frac{\alpha }{3})$ and the roughness increases as $O(t^{\frac{\alpha }{3}})$, during the coarsening dynamics with the random initial condition. That is to say, the coarsening rate of time fractional MBE model could be manipulated by the fractional order $\alpha$ as a power law proportional to $\alpha$. This is the first time in literature to report/discover such scaling correlation for the MBE model. It provides a potential application field for fractional differential equations to study anomalous coarsening. Besides, the numerical approximation strategy proposed in this paper can be readily applied to study many classes of time-fractional phase field models.