A block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition

ABSTRACT

In this article, a block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition is introduced and analysed. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence $O(\mathrm{\Delta }{t}^{1+\sigma /2}+h^2+k^2+{\sigma }^2)$ both for pressure and velocity are established on non-uniform rectangular grids, where $\mathrm{\Delta }t,h,k$ and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Keywords:

• Block-centred finite difference
• distributed-order differential equation
• stability
• priori estimates
• numerical experiments

2010 AMS Subject Classifications:

• 65M06
• 65M12
• 65M15
• 26A33

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper. The author X. Li thanks for the financial support from China Scholarship Council.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [Grant no. 11671233].