Compatible $L^2$ norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection

TL;DR

This paper establishes a novel $L^2$ norm error estimate for a variable-step L1 scheme applied to a time-fractional MBE model with slope selection, ensuring asymptotic compatibility and preservation of physical properties.

Contribution

It introduces the first asymptotically compatible $L^2$ error estimate for nonlinear subdiffusion problems with variable steps under a CSS-consistent condition.

Findings

The scheme maintains physical properties like volume conservation and energy dissipation.

Error estimates are compatible with classical schemes as fractional order approaches 1.

Numerical experiments confirm theoretical convergence and properties.

Abstract

The convergence of variable-step L1 scheme is studied for the time-fractional molecular beam epitaxy (MBE) model with slope selection.A novel asymptotically compatible $L^2$ norm error estimate of the variable-step L1 scheme is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that 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 $\epsilon\rightarrow 0^+$. To the best of our knowledge, it is the first time to establish such error estimate for nonlinear subdiffusion problems. The asymptotically compatible convergence means that the error estimate is compatible with that of backward Euler scheme for the classical MBE model as the fractional order $\alpha\rightarrow 1^-$. Just as the backward Euler scheme can maintain the physical properties of the MBE equation, the variable-step L1 scheme can also preserve the corresponding properties of the time-fractional MBE model, including the volume conservation, variational energy dissipation law and $L^2$ norm boundedness. Numerical experiments are presented to support our theoretical results.