Skip to main content
SpringerLink
Log in

Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A fourth-order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The \(L1\) discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The convergence order is \(\mathcal{O }(\tau ^{3-\alpha }+h^4)\) in the maximum norm, where \(\tau \) is the temporal grid size and \(h\) is the spatial grid size, respectively. In addition, a Crank–Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global convergence order of \(\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)\) is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank–Nicolson scheme are presented to show the effectiveness of the compact scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Chen, C.M., Liu, F.W., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, 886–897 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, C.M., Liu, F.W., Turner, I., Anh, V.: Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algorithm 54, 1–21 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C.M., Liu, F.W., Turner, I., Anh, V.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740–1760 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cui, M.R.: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phys. 231, 2621–2633 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deng, W.H.: Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227, 1510–1522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, R., Cao, W.R., Sun, Z.Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34, 2998–3007 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gao, G.H., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)

    Google Scholar 

  9. Jiang, H., Liu, F.W., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl. 64, 3377–3388 (2012)

    Article  MathSciNet  Google Scholar 

  10. Langlands, T., Henry, B.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, J., Sun, Z.Z., Zhao, X.: A three level linearized compact difference scheme for the Cahn-Hilliard equation. Sci. China Math. 55, 805–826 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Liang, J.S., Chen, Y.Q.: Hybrid symbolic and numerical simulation studies of time-fractional order wave-diffusion systems. Internat. J. Control 79, 1462–1470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, Y.M., Li, X.J., Xu, C.J.: Finite difference/spectral approximations for the fractional cable equation. Math. Comput. 275, 1369–1396 (2011)

    MathSciNet  Google Scholar 

  16. Liu, F.W., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)

    MATH  Google Scholar 

  19. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-diffusion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Momani, S., Odibat, Z., Erturk, V.S.: Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation. Phys. Lett. A 370, 379–387 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Murillo, J.Q., Yuste, S.B.: An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form. J. Comput. Nonlinear Dyn 6, 021014 (2011)

    Article  Google Scholar 

  22. Murillo, J.Q., Yuste, S.B.: On three explicit difference schemes for fractional diffusion and diffusion-wave equations. Phys. Scr. 136, 014025 (2009)

    Article  Google Scholar 

  23. Naber, M.: Distributed order fractional sub-diffusion. Fractals 12, 23–32 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  25. Povstenko, Y.: Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59, 593–605 (2010)

    Article  MATH  Google Scholar 

  26. Ren, J.C., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456–467 (2013)

    Article  MathSciNet  Google Scholar 

  27. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \(R^2\). J. Comput. Appl. Math. 193, 243–268 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sun, Z.Z.: Compact difference schemes for the Heat equation with Neumann boundary conditions. Numer. Methods Partial Differ. Equs. 25, 1320–1341 (2009)

    Article  MATH  Google Scholar 

  29. Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2nd edn. Science Press, Beijing (2012). (in Chinese)

    Google Scholar 

  30. Sun, Z.Z.: The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations. Science Press, Beijing (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank–Nicolson-type difference scheme for the subdiffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, Y.N., Sun, Z.Z.: Alternating direction implicit schemes for the two-dimensional sub-diffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhao, X., Sun, Z.Z.: A box-type scheme for the fractonal sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6061–6074 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhuang, P.H., Liu, F.W., Anh, V., Turner, I.: Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74, 6445–6467 (2009)

    Article  MathSciNet  Google Scholar 

  39. Zhuang, P.H., Liu, F.W., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)

    Google Scholar 

  40. Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods. Springer, New York (1995)

    MATH  Google Scholar 

  41. Peacemann Jr, D.W., Rachford, H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  Google Scholar 

  42. Douglas Jr, J.: Alternating direction method for three space variables. Numer. Math. 4, 41–63 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  43. Douglas Jr, J., Gunn, J.E.: A general formulation of alternating direction method, I. Parabolic and hyperbolic problem. Numer. Math. 6, 428–453 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  44. D’Yakonov, E.G.: Difference schemes of second-order accuracy with a splitting operator for parabolic equations without mixed partial derivatives. Zh. Vychisl. Mat. I. Mat. Fiz. 4, 935–941 (1964)

    Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referees for many constructive comments and suggestions which led to an improved presentation of this paper. The research is supported by National Natural Science Foundation of China (No. 11271068) and the Research and Innovation Project for College Graduates of Jiangsu Province (No. CXZZ11_0134) and Foundation for Key Teacher of Shangqiu Normal University (No.2012GGJS14).

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Jincheng Ren & Zhi-zhong Sun

  2. Department of Mathematics, Shangqiu Normal University, Henan, 476000, People’s Republic of China

    Jincheng Ren

Authors
  1. Jincheng Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-zhong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-zhong Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ren, J., Sun, Zz. Numerical Algorithm With High Spatial Accuracy for the Fractional Diffusion-Wave Equation With Neumann Boundary Conditions. J Sci Comput 56, 381–408 (2013). https://doi.org/10.1007/s10915-012-9681-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9681-9

Keywords

Search

Navigation