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

by   Yin Yang, et al.
Nanjing University of Aeronautics and Astronautics
South China Normal University
Xiangtan University
NetEase, Inc

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 ϵ→ 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 α→ 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.


Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model

A new discrete energy dissipation law of the variable-step fractional BD...

Energy stability and convergence of variable-step L1 scheme for the time fractional Swift-Hohenberg model

A fully implicit numerical scheme is established for solving the time fr...

Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model

The positive definiteness of discrete time-fractional derivatives is fun...

The variable-step L1 scheme preserving a compatible energy law for time-fractional Allen-Cahn equation

In this work, we revisit the adaptive L1 time-stepping scheme for solvin...

Wasserstein Gradient Flow Formulation of the Time-Fractional Fokker-Planck Equation

In this work, we investigate a variational formulation for a time-fracti...

Asymptotically compatible piecewise quadratic polynomial collocation for nonlocal model

In this paper, we propose and analyze piecewise quadratic polynomial col...

Please sign up or login with your details

Forgot password? Click here to reset