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 and its efficient, accurate, full discrete, linear numerical approximation. The numerical scheme utilizes the fast algorithm for the Caputo fractional derivative operator in time discretization and Fourier spectral method in spatial discretization. Refinement tests are conducted to verify the $2-\alpha$ order of time convergence, with $\alpha \in (0,1]$ the fractional order of derivative. Several numerical simulations are presented to demonstrate the accuracy and efficiency of our newly proposed scheme. By exploring the fast algorithm calculating the Caputo fractional derivative, our numerical scheme makes it practice for long time simulation of MBE coarsening, which is essential for MBE model in practice. With the proposed fractional MBE model, we observe that the scaling law for the energy decays as $O(t^{-\frac{\alpha }{3}})$ and the roughness increases as $O(t^{\frac{\alpha }{3}})$, during the coarsening dynamics with random initial condition. That is to say, the coarsening rate of MBE model could be manipulated by the fractional order $\alpha$, and it is linearly proportional to $\alpha$. This is the first time in literature to report/discover such scaling correlation. It provides a potential application field for fractional differential equations. Besides, the numerical approximation strategy proposed in this paper can be readily applied to study many classes of time-fractional and high dimensional phase field models.

References in corpus (1)

• A new class of efficient and robust energy stable schemes for gradient flows

Cited by in corpus (16)

• A highly efficient and accurate exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach for dissipative system
• Highly efficient schemes for time fractional Allen-Cahn equation using extended SAV approach
• Adaptive linear second-order energy stable schemes for time-fractional Allen-Cahn equation with volume constraint
• Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations
• A Non-uniform Time-stepping Convex Splitting Scheme for the Time-fractional Cahn-Hilliard Equation
• A decreasing upper bound of energy for time-fractional phase-field equations
• Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
• Compatible $L^2$ norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection
• Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model
• Time-Fractional Allen-Cahn Equations: Analysis and Numerical Methods
• Step-by-step solving schemes based on scalar auxiliary variable and invariant energy quadratization approaches for gradient flows
• The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing
• Efficient invariant energy quadratization and scalar auxiliary variable approaches without bounded below restriction for phase field models
• Energy stable schemes for gradient flows based on novel auxiliary variable with energy bounded above
• Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation
• The fast scalar auxiliary variable approach with unconditional energy stability for nonlocal Cahn-Hilliard equation