Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A FINITE DIFFERENCE/FINITE VOLUME METHOD FOR SOLVING THE FRACTIONAL DIFFUSION WAVE EQUATION

  • Sun, Yinan (Department of Mathematics Northeastern University) ;
  • Zhang, Tie (Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries Northeastern University)
  • Received : 2019.06.19
  • Accepted : 2021.02.15
  • Published : 2021.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: ∂βtu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂βtu by using an interpolation of derivative type. The truncation error of this formula is of O(△t2+δ-β)-order if function u(t) ∈ C2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t3-β) if u(t) ∈ C3 [0, T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H1-norm and L2-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.

Keywords

Acknowledgement

This work was supported by the State Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds, No. 2013ZCX02.

References

  1. P. L. Butzer and U. Westphal, An introduction to fractional calculus, in Applications of fractional calculus in physics, 1-85, World Sci. Publ., River Edge, NJ, 2000. https://doi.org/10.1142/9789812817747_0001
  2. L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003 (2003), no. 54, 3413-3442. https://doi.org/10.1155/S0161171203301486
  3. R. E. Ewing, T. Lin, and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal. 39 (2002), no. 6, 1865-1888. https://doi.org/10.1137/S0036142900368873
  4. G. Gao, Z. Sun, and H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys. 259 (2014), 33-50. https://doi.org/10.1016/j.jcp.2013.11.017
  5. Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math. 235 (2011), no. 11, 3285-3290. https://doi.org/10.1016/j.cam.2011.01.011
  6. B. Jin, R. Lazarov, and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), no. 1, 445-466. https://doi.org/10.1137/120873984
  7. S. Karaa, K. Mustapha, and A. K. Pani, Optimal error analysis of a FEM for fractional diffusion problems by energy arguments, J. Sci. Comput. 74 (2018), no. 1, 519-535. https://doi.org/10.1007/s10915-017-0450-7
  8. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  9. N. Kumar and M. Mehra, Collocation method for solving non-linear fractional optimal con trol problems by using Hermite scaling function with error estimates, Optimal Control, Appl. Meth. (2020). https://doi.org/10.1002/oca.2681
  10. Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533-1552. https://doi.org/10.1016/j.jcp.2007.02.001
  11. J. Lv and Y. Li, L2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math. 37 (2012), no. 3, 393-416. https://doi.org/10.1007/s10444-011-9215-2
  12. W. McLean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math. 105 (2007), no. 3, 481-510. https://doi.org/10.1007/s00211-006-0045-y
  13. V. Mehandiratta and M. Mehra, A difference scheme for the time-fractional diffusion equation on a metric star graph, Appl. Numer. Math. 158 (2020), 152-163. https://doi.org/10.1016/j.apnum.2020.07.022
  14. V. Mehandiratta, M. Mehra, and G. Leugering, An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems, Math. Methods Appl. Sci. 44 (2021), no. 4, 3195-3213. https://doi.org/10.1002/mma.6800
  15. K. S. Patel and M. Mehra, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, J. Comput. Appl. Math. 380 (2020), 112963, 15 pp. https://doi.org/10.1016/j.cam.2020.112963
  16. I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
  17. J. Ren, X. Long, S. Mao, and J. Zhang, Superconvergence of finite element approximations for the fractional diffusion-wave equation, J. Sci. Comput. 72 (2017), no. 3, 917-935. https://doi.org/10.1007/s10915-017-0385-z
  18. A. K. Singh and M. Mehra, Uncertainty quantification in fractional stochastic integro-differential equations using Legendre wavelet collocation method, Lecture Notes in Computer Science 12138 (2020), 58-71.
  19. M. Stynes, E. O'Riordan, and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057-1079. https://doi.org/10.1137/16M1082329
  20. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193-209. https://doi.org/10.1016/j.apnum.2005.03.003
  21. Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), 1-15. https://doi.org/10.1016/j.jcp.2014.08.012
  22. S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), no. 1, 264-274. https://doi.org/10.1016/j.jcp.2005.12.006
  23. F. Zeng, C. Li, F. Liu, and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput. 35 (2013), no. 6, A2976-A3000. https://doi.org/10.1137/130910865
  24. T. Zhang, Superconvergence of finite volume element method for elliptic problems, Adv. Comput. Math. 40 (2014), no. 2, 399-413. https://doi.org/10.1007/s10444-013-9313-4
  25. T. Zhang and Q. Guo, The finite difference/finite volume method for solving the fractional diffusion equation, J. Comput. Phys. 375 (2018), 120-134. https://doi.org/10.1016/j.jcp.2018.08.033
  26. Y. Zhao, Y. Zhang, D. Shi, F. Liu, and I. Turner, Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations, Appl. Math. Lett. 59 (2016), 38-47. https://doi.org/10.1016/j.aml.2016.03.005