Energy stability of a temporal variable-step difference scheme for time-fractional nonlinear fourth-order reaction–diffusion equation

Abstract

In this paper, we propose a nonuniform numerical formula for the Caputo fractional derivative at the half-grid based on the piecewise linear interpolation and construct a difference scheme for the time-fractional nonlinear fourth-order reaction–diffusion equation. By virtue of two discrete tools: the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, we obtain the positive definiteness of the discrete time-fractional derivative. Then, a discrete variational energy dissipation law of the proposed difference scheme is established for the time-fractional nonlinear fourth-order reaction–diffusion equation, which is asymptotically compatible with the associated energy dissipation law for the classical equation as the value of fractional order approaches to one. Numerical experiments demonstrate the effectiveness and the energy dissipation of the proposed difference scheme with an adaptive time-stepping strategy.

Keywords:

• Graded mesh
• fractional derivative
• energy dissipation law
• finite difference scheme

Mathematics Subject Classification (2010):

• 65M06
• 65M12

Disclosure statement

No potential conflict of interest was reported by the authors.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Nos. 12071073, 11701229). The first author would also like to thank the financial support from the China Postdoctoral Science Foundation (No. 2019M651634) and the Start-up Funding for High-level Scientific Research of Nanjing Institute of Technology (No. YKL201856).