A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions

Abstract

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the $\beta \to 1$, where $1-\beta (\in (0,1))$ is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

Keywords:

• novel compact ADI scheme
• subdiffusion
• modified fractional diffusion equation
• stability

2010 AMS Subject Classifications:

• 35R11
• 65M06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was partially supported by the Natural Science Foundation of China under [grant number 11372170]; the Key Program of Shanghai Municipal Education Commission under [grant number 12ZZ084]; and the grant of ‘The First-class Discipline of Universities in Shanghai’.