Efficient compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients

ABSTRACT

This paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients. Based on some new techniques coupled with the $L2$-$1_{\sigma }$ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods.

KEYWORDS:

• Fractional convection–reaction–diffusion equation
• variable coefficient
• compact finite difference method
• stability and high-order convergence
• error estimate

MATHEMATICAL SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15
• 35R11

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Science and Technology Commission of Shanghai Municipality (STCSM) (No. 13dz2260400).