Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

Abstract

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2−α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.

Keywords:

• reaction–subdiffusion equation
• compact difference method
• Caputo derivative
• Fourier method
• stability
• convergence
• Neumann boundary value condition

2010 AMS Subject Classifications::

• 65M06
• 65M12

Acknowledgements

The work was partially supported by the Natural Science Foundation of China under Grant No. 11372170, the Key Program of Shanghai Municipal Education Commission under Grant No. 12ZZ084 and the grant of ‘The First-class Discipline of Universities in Shanghai’.