An extrapolated finite difference method for two-dimensional fractional boundary value problems with non-smooth solution

Abstract

In this paper, the well-known shifted Grünwald–Letnikov formula is revisited to solve the two-dimensional fractional boundary value problems (FBVPs) with non-smooth solution. Both stability analysis and error estimates of the scheme are carried out in the maximum norm. An improved algorithm is presented based on the extrapolation technique to obtain high-order accuracy for the problems with low regularity. Various numerical experiments are given to support the theoretical finding and verify the effectiveness of the improved algorithm. It is illustrated that, by using the proposed algorithm, both accuracy and convergence rate can be significantly improved for solving the two-dimensional FBVPs as well as fractional diffusion equations with non-smooth solution. Especially, the convergence rate of corrected numerical solutions even can reach second-order in the case of solving problems involved one-sided fractional derivatives.

Keywords:

• Numerical correction
• spatial fractional derivative
• low regularity
• error estimate in maximum norm
• convergence rate

AMS Subject Classifications:

• 26A33
• 65M06
• 65M12
• 65M55
• 65T50

Disclosure statement

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

Additional information

Funding

The research is partially supported by the National Natural Science Foundation of China [grant number 11671083] and the Fundamental Research Funds for the Central Universities.