Skip to main content
SpringerLink
Log in

Finite difference schemes for the two-dimensional multi-term time-fractional diffusion equations with variable coefficients

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Two implicit finite difference schemes for solving the two-dimensional multi-term time-fractional diffusion equation with variable coefficients are considered in this paper. The orders of the Riemann–Liouville fractional time derivatives acting on the spatial derivatives can be different in various spatial directions. By integrating the original partial differential equation with time variable first, and the second-order spatial derivatives are approximated by the central difference quotients, then the fully discrete finite difference scheme can be obtained after the right rectangular quadrature formulae are used to approximate the resulting time integrals. The convergence analysis is given by the energy method, showing that the difference scheme is first-order accurate in time and second order in space. Based on a second-order approximation of the Riemann–Liouville fractional derivatives using the weighted and shifted Grünwald difference operator, we present the Crank–Nicolson scheme and prove it is second-order accurate both in time and space. Numerical results are provided to verify the accuracy and efficiency of the two proposed algorithms. Numerical schemes and theoretical analysis can be generalized for the three-dimensional problems.

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
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Cao JX, Song GJ, Wang J, Shi QH, Sun SJ (2019) Blow-up and global solutions for a class of time fractional nonlinear reaction-diffusion equation with weakly spatial source. Appl Math Lett 91:201–206

    MathSciNet  MATH  Google Scholar 

  • Chen C, Liu F, Anh V, Turner V (2011) Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math Comput 81:345–366

    MathSciNet  MATH  Google Scholar 

  • Chen CM, Liu F, Burrage K (2008) Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation. Appl Math Comput 198:754–769

    MathSciNet  MATH  Google Scholar 

  • Chen S, Liu F, Jiang X, Turner I, Anh V (2015) A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients. Appl Math Comp 257:591–601

    MathSciNet  MATH  Google Scholar 

  • Chen S, Liu F, Jiang X, Turner I, Burrage K (2016) Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients. SIAM J Numer Anal 54:606–624

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Cheng BR, Guo ZH, Wang DL (2018) Dissipativity of semilinear time fractional subdiffusion equations and numerical approximations. Appl Math Lett 86:276–283

    MathSciNet  MATH  Google Scholar 

  • Cui MR (2009) Compact finite difference method for the fractional diffusion equation. J Comput Phys 228:7792–7804

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Cui MR (2013) Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer Algor 62:383–409

    MathSciNet  MATH  Google Scholar 

  • Cui MR (2019) Finite difference schemes for the variable coefficients single and multi-term time-fractional diffusion equations with non-smooth solutions on graded and uniform meshes. Numer Math Theor Meth Appl 12:845–866

    MathSciNet  MATH  Google Scholar 

  • Dehghan M, Abbaszadeh M, Mohebbi A (2016) Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation. Appl Math Model 40:3635–3654

    MathSciNet  MATH  Google Scholar 

  • Duan B, Jin B, Lazarov R, Pasciak J, Zhou Z (2018) Space-time petrov-Galerkin FEM for fractional diffusion problems. Comput Methods Appl Math 18:1–20

    MathSciNet  MATH  Google Scholar 

  • Edwards JT, Ford NJ, Simpson AC (2002) The numerical solution of linear multi-term fractional differential equations systems of equations. J Comput Appl Math 148:401–418

    MathSciNet  MATH  Google Scholar 

  • Gao GH, Alikhanov AA, Sun ZZ (2017) The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J Sci Comput 73:93–121

    MathSciNet  MATH  Google Scholar 

  • Gao GH, Sun ZZ (2011) A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 230:586–595

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Jin B, Lazarov R, Liu Y, Zhou Z (2015) The Galerkin finite element method for a multi-term time-fractional diffusion equation. J Comput Phys 281:825–843

    MathSciNet  MATH  Google Scholar 

  • Langlands TAM, Henry BI (2000) The accuracy and stability of an implicit solution method for the fractional diffusion equation. J Comput Phys 205:719–736

    MathSciNet  MATH  Google Scholar 

  • Li Z, Liu Y, Yamamoto M (2015) Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl Math Comput 257:381–397

    MathSciNet  MATH  Google Scholar 

  • Lin YM, Xu CJ (2007) Finite difference/spectral approximation for the time fractional diffusion equations. J Comput Phys 225:1533–1552

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Liu FW, Zhuang PH, Liu QX (2015) Numerical methods for fractional partial differential equations with applications. In: Information and computational sciences series, vol 74. Science Press, Beijing

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Mohebbi A, Abbaszadeh M, Dehghan M (2013) A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term. J Comput Phys 240:36–48

    MathSciNet  MATH  Google Scholar 

  • Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York

    MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  • Qin S, Liu F, Turner I, Vegh V, Yu Q, Yang Q (2017) Multi-term time-fractional Bloch equations and application in magnetic resonance imaging. J Comput Appl Math 319:308–319

    MathSciNet  MATH  Google Scholar 

  • Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations. Springer, Berlin

    MATH  Google Scholar 

  • Reutskiy SYu (2017) A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients. Appl Math Model 45:238–254

    MathSciNet  MATH  Google Scholar 

  • Samarskii AA, Andreev VB (1976) Difference methods for elliptic equation. Nauka, Moscow

    MATH  Google Scholar 

  • Schumer R, Benson DA, Benson MM, Baeumer B (2003) Fractal mobile/immobile solute transport. Water Resour Res 39:1296. https://doi.org/10.1029/2003WR002141

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Sun ZZ (2012) Numerical solutions of partial differential equations, 2nd edn. Sciences Press, Beijing

    Google Scholar 

  • Sun ZZ, Gao GH (2021) Finite difference method for fractional differential equations, 2nd edn. Information and computational sciences series, vol 87. Science Press, Beijing

  • Thomas JW (1995) Numerical partial differential equations finite difference methods. In: Texts in applied mathematics, vol 22. Springer-Verlag

  • Tian WY, Zhou H, Deng WH (2015) A class of second order difference approximations for solving space fractional diffusion equations. Math Comput 84:1703–1727

    MathSciNet  MATH  Google Scholar 

  • Wang H, Du N (2013) A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations. J Comput Phys 240:49–57

    MathSciNet  MATH  Google Scholar 

  • Wang ZB, Vong S (2014) Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J Comput Phys 277:1–15

    MathSciNet  MATH  Google Scholar 

  • Wang H, Wang K, Sircar T (2010) A direct \(O(\mathit{Nlog^2 N)}\) finite difference method for fractional diffusion equations. J Comput Phys 229:8095–8104

    MathSciNet  MATH  Google Scholar 

  • Yuste SB, Acedo L (2005) An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J Numer Anal 42:1862–1874

    MathSciNet  MATH  Google Scholar 

  • Zeng FH, Li CP, Liu FW, Turner I (2013) The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J Sci Comput 35:A2976–A3000

    MathSciNet  MATH  Google Scholar 

  • Zeng FH, Li CP, Liu FW, Turner I (2015) Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J Sci Comput 37:A55–A78

    MathSciNet  MATH  Google Scholar 

  • Zeng F, Zhang Z, Karniadakis GE (2016) Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. J Comput Phys 307:15–33

    MathSciNet  MATH  Google Scholar 

  • Zhang YN, Sun ZZ (2011) Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J Comput Phys 230:8713–8728

    MathSciNet  MATH  Google Scholar 

  • Zhang YN, Sun ZZ, Zhao X (2012) Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J Numer Anal 50:1535–1555

    MathSciNet  MATH  Google Scholar 

  • Zhao YM, Zhang YD, Liu F, Turner I, Shi DY (2016) Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation. Appl Math Model 40:8810–8825

    MathSciNet  MATH  Google Scholar 

  • Zhao Y, Zhang Y, Liu F, Turner I, Tang Y, Anh V (2017) Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations. Comput Math Appl 73:1087–1099

    MathSciNet  MATH  Google Scholar 

  • Zheng M, Liu F, Anh V, Turner I (2016) A high-order spectral method for the multi-term time-fractional diffusion equations. Appl Math Model 40:4970–4985

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Zhuang P, Liu F, Anh V, Turner I (2009) Stability and convergence of an implicit numerical method for the nonlinear fractional reaction-subdiffusion process. IMA J Appl Math 74:645–667

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous referees and the associate editor for giving many valuable and helpful comments to improve this work greatly. The author wants to express his thanks to Prof. Hong Wang, University of South Carolina, USA, for suggestion on considering this problem and many instructions and kind help.

Author information

Authors and Affiliations

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

    Mingrong Cui

Authors
  1. Mingrong Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingrong Cui.

Additional information

Communicated by Frederic Valentin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by Natural Science Foundation of Shandong Province (Grant No. ZR2014AM013), NSFC (Grant No. 11471194), NSAF (Grant No. U1430101)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cui, M. Finite difference schemes for the two-dimensional multi-term time-fractional diffusion equations with variable coefficients. Comp. Appl. Math. 40, 167 (2021). https://doi.org/10.1007/s40314-021-01551-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01551-1

Keywords

Mathematics Subject Classification

Search

Navigation