, Volume 54, Issue 3, pp 733–768 | Cite as

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

  • Yuan-Ming WangEmail author


This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.


Fractional mobile/immobile convection–diffusion equation Compact finite difference method Shifted Grünwald formula Stability and convergence Richardson extrapolation 

Mathematics Subject Classification

65M06 65M12 65M15 35R11 



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


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© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghaiPeople’s Republic of China

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