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A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations

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Abstract

In this paper, we investigate the well-posedness and solution regularity of a variable-order time-fractional diffusion equation, which is often used to model the solute transport in complex porous media where the micro-structure of the porous media may changes over time. We show that the variable-order time-fractional diffusion equations have flexible abilities to eliminate the nonphysical singularity of the solutions to their constant-order analogues. We also present a finite volume approximation and provide its stability and convergence analysis in a weighted discrete norm. Furthermore, we develop an efficient parallel-in-time procedure to improve the computational efficiency of the variable-order time-fractional diffusion equations. Numerical experiments are performed to confirm the theoretical results and to demonstrate the effectiveness and efficiency of the parallel-in-time method.

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References

  1. Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2009)

    Article  MathSciNet  Google Scholar 

  2. Ervin, V., Heuer, N., Roop, J.: Regularity of the solution to 1-D fractional order diffusion equations. Math. Comput. 87, 2273–2294 (2018)

    Article  MathSciNet  Google Scholar 

  3. Evans, L.C.: Graduate Studies in Mathematics. Partial Differential Equations, vol. 19. American Mathematical Society, Rhode Island (1998)

    Google Scholar 

  4. Fu, H., Liu, H., Wang, H.: A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation. J. Comput. Phys. 388, 316–334 (2019)

    Article  MathSciNet  Google Scholar 

  5. Fu, H., Wang, H.: A preconditioned fast parareal finite difference method for space-time fractional partial differential equation. J. Sci. Comput. 78, 1724–1743 (2019)

    Article  MathSciNet  Google Scholar 

  6. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  7. Jia, J., Zheng, X., Fu, H., Dai, P., Wang, H.: A fast method for variable-order space-fractional diffusion equations. Numer. Algorithms (2020). https://doi.org/10.1007/s11075-020-00875-z

    Article  Google Scholar 

  8. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21(3), 650–678 (2017)

    Article  MathSciNet  Google Scholar 

  9. Jiang, W., Liu, N.: A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model. Appl. Numer. Math. 119, 18–32 (2017)

    Article  MathSciNet  Google Scholar 

  10. Ke, R., Ng, M.K., Sun, H.W.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys 303(C), 203–211 (2015)

    Article  MathSciNet  Google Scholar 

  11. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)

    Article  MathSciNet  Google Scholar 

  12. Li, C., Ding, H.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38, 3802–3821 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Liu, Y., Du, Y., Li, H., Liu, F., Wang, Y.: Some second-order \(\theta \) schemes combined with finite element method for nonlinear fractional cable equation. Numer. Algorithms 80(2), 533–555 (2019)

    Article  MathSciNet  Google Scholar 

  15. Liu, Z., Li, X.: A Crank-Nicolson difference scheme for the time variable fractional mobile-immobile advection-dispersion equation. J. Appl. Math. Comput. 56(1–2), 391–410 (2018)

    Article  MathSciNet  Google Scholar 

  16. Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64(10), 2990–3007 (2012)

    Article  MathSciNet  Google Scholar 

  17. Ma, H., Yang, Y.: Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model. East Asian J. Appl. Math. 6(3), 337–352 (2016)

    Article  MathSciNet  Google Scholar 

  18. Meerschaert M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics (2011)

  19. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  20. Podlubny, I.: Fractional Differential Equations. Academic Press, Cambridge (1999)

    MATH  Google Scholar 

  21. 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 

  22. Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39(10), 1296 (2003)

    Article  Google Scholar 

  23. Shao, J.: New integral inequalities with weakly singular kernel for discontinuous functions and their applications to impulsive fractional differential systems. J. Appl. Math. (2014). https://doi.org/10.1155/2014/252946

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Sun, H.G., Chang, A., Zhang, Y., Chen, W.: A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22(1), 27–59 (2019)

    Article  MathSciNet  Google Scholar 

  26. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    Article  MathSciNet  Google Scholar 

  27. Umarov, S.R., Steinberg, S.T.: Variable order differential equations with piecewise constant order function and diffusion with changing modes. Z. Anal. Anwend. 28, 431–450 (2009)

    Article  MathSciNet  Google Scholar 

  28. Wang, H., Zheng, X.: Wellposedness and regularity of the variable-order time-fractional diffusion equations. J. Math. Anal. Appl. 475, 1778–1802 (2019)

    Article  MathSciNet  Google Scholar 

  29. Wang, H., Zheng, X.: Analysis and numerical solution of a nonlinear variable-order fractional differential equation. Adv. Comput. Math. 45, 2647–2675 (2019)

    Article  MathSciNet  Google Scholar 

  30. Wu, S., Zhou, T.: Parareal algorithms with local time-integrators for time fractional differential equations. J. Comput. Phys. 358, 135–149 (2018)

    Article  MathSciNet  Google Scholar 

  31. Xian, Y., Jin, M., Zhan, H., Liu, Y.: Reactive transport of nutrients and bioclogging during dynamic disconnection process of stream and groundwater. Water Resour. Res. 55, 3882–3903 (2019)

    Article  Google Scholar 

  32. Xu, Q., Hesthaven, J.S., Chen, F.: A parareal method for time-fractional differential equations. J. Comput. Phys. 293, 173–183 (2015)

    Article  MathSciNet  Google Scholar 

  33. Yin, B., Liu, Y., Li, H., He, S.: Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions. J. Comput. Phys. 379, 351–372 (2019)

    Article  MathSciNet  Google Scholar 

  34. Zeng, F., Zhang, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. SIAM Sci. Comput. 37, A2710–A2732 (2015)

    Article  MathSciNet  Google Scholar 

  35. Zhao, X., Sun, Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)

    Article  MathSciNet  Google Scholar 

  36. 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(5), 693–701 (2013)

    Article  MathSciNet  Google Scholar 

  37. Zheng, X., Wang, H.: Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J. Numer. Anal. (2020). https://doi.org/10.1093/imanum/draa013

    Article  Google Scholar 

  38. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Numer. Anal. 47, 1760–1781 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their sincere thanks to the editor and referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was funded by the National Natural Science Foundation of China under Grants 91630207, 11471194, 11971272, the OSD/ARO MURI Grant W911NF-15-1-0562 and the National Science Foundation under Grant DMS-1620194.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Huan Liu & Aijie Cheng

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang

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  1. Huan Liu
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  2. Aijie Cheng
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  3. Hong Wang
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Correspondence to Aijie Cheng.

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Liu, H., Cheng, A. & Wang, H. A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations. J Sci Comput 85, 19 (2020). https://doi.org/10.1007/s10915-020-01321-x

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