Robust and stable schemes for time fractional molecular beam epitaxial growth model using SAV approach

Abstract

In this paper, we are interested in the time fractional molecular beam epitaxial growth models and its numerical solutions. Basically, two types of numerical schemes are proposed and analyzed: one is based on the Crank–Nicolson formula while the other is a kind of backward differentiation formula. The former is proved to be unconditional energy stable with a modified energy on the general temporal mesh, including the graded mesh and the uniform mesh. However, A drawback is its sensitivity to the coefficient and the time step size. The other, i.e., the backward differentiation based scheme, is more robust in term of this sensitivity. Its unconditional stability with respect to the modified energy is proved only for the uniform mesh. The key of these results is the use of a new approach to the time fractional derivative. The idea is to first split the time fractional derivative into the local part and the history part, then treat the two parts in different ways. This splitting allows to derive a nonlocal free energy dispassion for the continuous solution, and makes the construction of stable numerical schemes much easier. It is found that the well known L1, L1–CN, L1+–CN and L2 formula with the sum-of-exponentials approach for the local derivative term, combined with the extended auxiliary approach for the nonlinear and history terms result in unconditional stable schemes, which satisfy the modified energy dispassion at the discrete level. Finally, a series of numerical examples are provided to verify the efficiency of the proposed schemes.