Two energy stable variable-step L1 schemes for the time-fractional MBE model without slope selection

A variable-step L1 scheme is proposed for the time-fractional molecular beam epitaxy model without slope selection. By taking advantage of the convex splitting of nonlinear bulk, a sharp L2 norm error estimate is established under a convergence–solvability–stability (CSS) consistent time-step constraint, that is, the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter ϵ→0+. By using the discrete gradient structure of L1 formula, we build up a discrete variational energy dissipation law, which is asymptotically compatible with the classical energy dissipation law, as the fractional order α→1−. The stability and convergence of the stabilized convex splitting scheme are also investigated. The resulting scheme is shown to unconditionally preserve the variational energy dissipation law by adding a (2−α)-order stabilization term like τn1−α(ϕn−ϕn−1). Numerical experiments are presented to support our theoretical results.