Journal of Scientific Computing

, Volume 80, Issue 1, pp 1–25 | Cite as

Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations

  • Hong-lin Liao
  • Yonggui Yan
  • Jiwei ZhangEmail author


A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost \(O(MN^2)\) and storage O(MN) for the standard L1 formula to \(O(MN\log N)\) and \(O(M\log N)\), respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time \(t=0\), and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete \(H^2\) energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.


Semilinear subdiffusion equation Two-level L1 formula Discrete fractional Grönwall inequality Discrete \(H^2\) energy method Unconditional convergence 



The authors gratefully thank Professor Martin Stynes for his valuable discussions and fruitful suggestions during the preparation of this paper. Hong-lin Liao would also thanks for the hospitality of Beijing CSRC during the period of his visit.


  1. 1.
    Baffet, D., Hesthaven, J.S.: A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal. 55(2), 496–520 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15. Cambridge University Press, Cambridge (2004)zbMATHCrossRefGoogle Scholar
  3. 3.
    Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1–23 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Li, B., Wang, J., Sun, W.: The stability and convergence of fully discrete Galerkin FEMs for porous medium flows. Commun. Comput. Phys. 15, 1141–1158 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316, 614–631 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Li, J.: A fast time stepping method for evaluating fractional integrals. SIAM J. Sci. Comput. 31, 4696–4714 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Liao, H.-L., Sun, Z.Z., Shi, H.S.: Error estimate of fourth-order compact scheme for solving linear Schrödinger equations. SIAM. J. Numer. Anal. 47(6), 4381–4401 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Liao, H.-L., Sun, Z.Z., Shi, H.S.: Maximum norm error analysis of explicit schemes for two-dimensional nonlinear Schrödinger equations. Sci. China Math. 40(9), 827–842 (2010). (in Chinese) Google Scholar
  13. 13.
    Liao, H.-L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods PDEs 26, 37–60 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Liao, H.-L., Li, D., Zhang, J.: Sharp error estimate of nonuniform L1 formula for linear reaction–subdiffusion equations. SIAM J. Numer. Anal. 56(2), 1112–1133 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Liao, H.-L., McLean, W., Zhang, J.: A discrete Grönwall inequality with application to numerical schemes for reaction–subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218–237 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    McLean, W.: Fast summation by interval clustering for an evolution equation with memory. SIAM J. Sci. Comput. 34, A3039–A3056 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Mustapha, K., McLean, W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78(268), 1975–1995 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Mustapha, K., Mustapha, H.: A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel. IMA J. Numer. Anal. 30(2), 555–578 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Xu, Q., Hesthaven, J.S., Chen, F.: A parareal method for time-fractional differential equations. J. Comput. Phys. 293(C), 173–183 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Yan, Y., Sun, Z.Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22, 1028–1048 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhang, Y.N., Sun, Z.Z., Liao, H.-L.: Finite difference methods for the time fractional diffusion equation on nonuniform meshes. J. Comput. Phys. 265, 195–210 (2014)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.Beijing Computational Science Research Center (CSRC)BeijingPeople’s Republic of China
  3. 3.School of Mathematics and Statistics, and Hubei Key Laboratory of Computational ScienceWuhan UniversityWuhanPeople’s Republic of China

Personalised recommendations