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A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction–subdiffusion problems with a weak initial singularity

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Abstract

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete \(L^{2}\)-norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.

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References

  1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  Google Scholar 

  2. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012)

    Book  Google Scholar 

  3. Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)

    Article  MathSciNet  Google Scholar 

  4. Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)

    Article  MathSciNet  Google Scholar 

  5. Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79, 624–647 (2019)

    Article  MathSciNet  Google Scholar 

  6. Cui, M.: Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficients. J. Comput. Phys. 280, 143–163 (2015)

    Article  MathSciNet  Google Scholar 

  7. Cui, M.: Combined compact difference scheme for the time fractional convection–diffusion equation with variable coefficients. Appl. Math. Comput. 246, 464–473 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Gracia, J.L., O’Riordan, E., Stynes, M.: Convergence in positive time for a finite difference method applied to a fractional convection–diffusion problem. Comput. Methods Appl. Math. 18, 33–42 (2018)

    Article  MathSciNet  Google Scholar 

  9. Guo, B.-Y., Wang, Y.-M.: An almost monotone approximation for a nonlinear two-point boundary value problem. Adv. Comput. Math. 8, 65–96 (1998)

    Article  MathSciNet  Google Scholar 

  10. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  Google Scholar 

  11. Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Dover Publications Inc, New York (1994)

    MATH  Google Scholar 

  12. Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561–582 (2015)

    Article  MathSciNet  Google Scholar 

  13. Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)

    Article  MathSciNet  Google Scholar 

  15. Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88, 2135–2155 (2019)

    Article  MathSciNet  Google Scholar 

  16. Lai, M., Tseng, Y.: A fast iterative solver for the variable coefficient diffusion equation on a disk. J. Comput. Phys. 208, 196–205 (2005)

    Article  MathSciNet  Google Scholar 

  17. Liao, H.-L., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction–subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)

    Article  MathSciNet  Google Scholar 

  18. Liao, H.-L., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction–sudiffusion problem. arXiv:1803.09873v2

  19. Liao, H.-L., Mclean, W., Zhang, J.: A discrete grönwall inequality with application to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218–237 (2019)

    Article  MathSciNet  Google Scholar 

  20. Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, 141–160 (2012)

    Article  MathSciNet  Google Scholar 

  21. Lv, C., Xu, C.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)

    Article  MathSciNet  Google Scholar 

  22. Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51, 650–682 (2012)

    Article  MathSciNet  Google Scholar 

  23. McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)

    Article  MathSciNet  Google Scholar 

  24. Mustapha, K.: FEM for time-fractional diffusion equations, novel optimal error analyses. Math. Comput. 87, 2259–2272 (2018)

    Article  MathSciNet  Google Scholar 

  25. Mustapha, K., Abdallah, B., Furati, K.M., Nour, M.: A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients. Numer. Algor. 73, 517–534 (2016)

    Article  MathSciNet  Google Scholar 

  26. Mustapha, K., Almutawa, J.: A finite difference method for an anomalous sub-diffusion equation, theory and applications. Numer. Algorithm 61, 525–543 (2012)

    Article  MathSciNet  Google Scholar 

  27. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  28. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  29. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  30. Sjögreen, B., Petersson, N.A.: A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation. J. Sci. Comput. 52, 17–48 (2012)

    Article  MathSciNet  Google Scholar 

  31. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)

    Article  MathSciNet  Google Scholar 

  32. Wang, Y.-M., Ren, L.: High-order compact difference methods for Caputo-type variable coefficient fractional sub-diffusion equations in conservative form. J. Sci. Comput. 76, 1007–1043 (2018)

    Article  MathSciNet  Google Scholar 

  33. Wang, Y.-M., Ren, L.: Efficient compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients. Int. J. Comput. Math. 96, 264–297 (2019)

    Article  MathSciNet  Google Scholar 

  34. Wang, Y.-M., Ren, L.: A high-order \(L2\)-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients. Appl. Math. Comput. 342, 71–93 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56, 210–227 (2018)

    Article  MathSciNet  Google Scholar 

  36. Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model. 38, 3848–3859 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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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  1. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, 200241, People’s Republic of China

    Yuan-Ming Wang

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  1. Yuan-Ming Wang
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Correspondence to Yuan-Ming Wang.

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Communicated by Axel Målqvist.

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This work was supported in part by Science and Technology Commission of Shanghai Municipality (STCSM) (No. 18dz2271000).

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Wang, YM. A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction–subdiffusion problems with a weak initial singularity. Bit Numer Math 61, 1023–1059 (2021). https://doi.org/10.1007/s10543-020-00841-0

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  • DOI: https://doi.org/10.1007/s10543-020-00841-0

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