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A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model

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Abstract

In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of L1 discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations; second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence in L2-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system.

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References

  1. Atangana, A., Baleanu, D.: Numerical solution of a kind of fractional parabolic equations via two difference schemes. In: Abstract and Applied Analysis, Hindawi Publishing Corporation (2013). https://doi.org/10.1155/2013/828764

  2. Chen, H., Xu, D.: A second-order fully discrete difference scheme for a nonlinear partial integro-differential equation (in Chinese). J. Sys. Sci. Math. Scis. 28, 51–70 (2008)

    MATH  Google Scholar 

  3. Chen, H., Gan, S., Xu, D., Liu, Q.: A second-order BDF compact difference scheme for fractional-order Volterra equations. Int. J. Computer Math. 93, 1140–1154 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Chen, H., Xu, D., Peng, Y.: An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation. Int. J. Comput. Math. 92, 2178–2197 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Deng, W., Hesthaven, J.S.: Local discontinuous Galerkin methods for fractional diffusion equations. ESAIM: M2AN. 47, 1186–1845 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Hu, S., Qiu, W., Chen, H.: A backward Euler difference scheme for the integro-differential equations with the multi-term kernels. Int. J. Comput Math. https://doi.org/10.1080/00207160.2019.1613529 (2019)

  8. Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comp. Phys. 316, 614–631 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Li, C., Zhao, Z., Chen, Y.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Li, D., Zhang, C., Ran, M.: A linear finite difference scheme for generalized time fractional Burgers equation. Appl. Math. Modelling. 40, 6096–6081 (2016)

    MathSciNet  Google Scholar 

  12. Li, X., Rui, H.: A two-grid block-centered finite difference method for the nonlinear time-fractional parabolic equation. J. Sci. Comput. 72, 863–891 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Lin, Y., Li, X., Xu, C.: Finite difference/spectral approximations for the fractional cable equation. Math. Comput. 80, 1369–1396 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64, 2990–3007 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38, 3871–3878 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Liu, Q., Liu, F., Turner, I., Anh, V., Gu, Y.: A RBF meshless approach for modeling a fractal mobile/immobile transport model. Appl. Math. Comput. 226, 336–347 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Liu, Y., Du, Y., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70, 573–591 (2015)

    MathSciNet  Google Scholar 

  19. Liu, Y., Du, Y., Li, H., Li, J., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative. Comput. Math. Appl. 70, 2474–2492 (2015)

    MathSciNet  Google Scholar 

  20. Liu, Y., Du, Y., Li, H., Wang, J.: A two-grid finite element approximation for a nonlinear time-fractional Cable equation. Nonlinear. Dyn. 85, 2535–2548 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Liu, Y., Yu, Z., Li, H., Liu, F., Wang, J.: Time two-mesh algorithm combined with finite element method for time fractional water wave model. Int. J. Heat Mass Transf. 120, 1132–1145 (2018)

    Google Scholar 

  22. Liu, Z., Cheng, A., Li, X.: A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative. Int. J. Comput. Math. 95, 396–411 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Lopez-Marcos, J.C.: A difference scheme for a nonlinear partial integro-differential equation. SIAM J. Numer. Anal. 27, 20–31 (1990)

    MathSciNet  MATH  Google Scholar 

  24. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  25. Qiao, L., Xu, D.: Compact alternating direction implicit scheme for integro-differential equations of parabolic type. J. Sci Comput. https://doi.org/10.1007/s10915-017-0630-5 (2018)

  26. Qiu, W., Chen, H., Zheng, X.: An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations. Math. Comput Simul. https://doi.org/10.1016/j.matcom.2019.05.017 (2019)

  27. Rui, H., Liu, W.: A two-grid block-centered finite difference method for Darcy-Forchheimer flow in porous media. SIAM J. Numer. Anal. 53, 1941–1962 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. https://doi.org/10.1029/2003WR002141 (2003)

  29. Sloan, I.H., Thomee, V.: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer. Anal. 23, 1052–1061 (1986)

    MathSciNet  MATH  Google Scholar 

  30. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 11, 309–319 (1993)

    MathSciNet  MATH  Google Scholar 

  32. Wei, L., He, Y.: Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38, 1511–1522 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    MathSciNet  MATH  Google Scholar 

  34. Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66, 693–701 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Zhang, H., Liu, F., Anh, V.: Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl. Math. Comput. 217, 2534–2545 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Zhou, J., Xu, D., Chen, H.: A weak Galerkin finite element method for multi-term time-fractional diffusion equations. East Asian J. Appl. Math. 8, 181–193 (2018)

    MathSciNet  Google Scholar 

  38. Zhou, J., Xu, D., Dai, X.: Weak Galerkin finite element method for the parabolic integro-differential equation with weakly singular kernel, Comput. Appl Math. https://doi.org/10.1007/s40314-019-0807-7 (2019)

  39. Zhuang, P., Liu, F., Turner, I., Gu, Y.: Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation. Appl. Math. Model. 38, 3860–3870 (2014)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the reviewers for their helpful and valuable suggestions/comments in the improvement of this paper.

Funding

This study is supported by the National Natural Science Foundation of China (No.11671131) and the Construct Program of the Key Discipline in Hunan Province, Performance Computing and Stochastic Information Processing (Ministry of Education of China).

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Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Wenlin Qiu, Da Xu, Jing Guo & Jun Zhou

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  1. Wenlin Qiu
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  2. Da Xu
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  3. Jing Guo
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  4. Jun Zhou
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Correspondence to Wenlin Qiu.

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Qiu, W., Xu, D., Guo, J. et al. A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. Numer Algor 85, 39–58 (2020). https://doi.org/10.1007/s11075-019-00801-y

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