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Journal of Scientific Computing

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01-05-2023

Global-in-Time \(H^1\)-Stability of L2-1\(_\sigma \) Method on General Nonuniform Meshes for Subdiffusion Equation

Authors: Chaoyu Quan, Xu Wu

Published in: Journal of Scientific Computing | Issue 2/2023

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Abstract

In this work the L2-1\(_\sigma \) method on general nonuniform meshes is studied for the subdiffusion equation. When the time step ratio is no less than 0.475329, a bilinear form associated with the L2-1\(_\sigma \) fractional-derivative operator is proved to be positive semidefinite and a new global-in-time \(H^1\)-stability of L2-1\(_\sigma \) schemes is then derived under simple assumptions on the initial condition and the source term. In addition, the sharp \(L^2\)-norm convergence is proved under the constraint that the time step ratio is no less than 0.475329.

previous article Numerical Analysis of the Nonuniform Fast L1 Formula for Nonlinear Time–Space Fractional Parabolic Equations

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Al-Maskari, M., Karaa, S.: The time-fractional Cahn-Hilliard equation: analysis and approximation. IMA J. Numer. Anal. 42(2), 1831–1865 (2022)MathSciNetCrossRefMATH

Anatoly, A., Alikhanov, A.: new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)MathSciNetCrossRefMATH

Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79(1), 624–647 (2019)MathSciNetCrossRefMATH

Chen, H., Stynes, M.: Blow-up of error estimates in time-fractional initial-boundary value problems. IMA J. Numer. Anal. 41(2), 974–997 (2021)MathSciNetCrossRefMATH

Gao, G., Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)MathSciNetCrossRefMATH

Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29(1), 129–143 (2002)MathSciNetCrossRefMATH

Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39(6), A3129–A3152 (2017)MathSciNetCrossRefMATH

Jin, B., Li, B.: Subdiffusion with time-dependent coefficients: improved regularity and second-order time stepping. Numer. Math. 145(4), 883–913 (2020)MathSciNetCrossRefMATH

Karaa, S.: Positivity of discrete time-fractional operators with applications to phase-field equations. SIAM J. Numer. Anal. 59(4), 2040–2053 (2021)MathSciNetCrossRefMATH

10.

Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88(319), 2135–2155 (2019)MathSciNetCrossRefMATH

11.

Kopteva, N.: Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem. Math. Comput. 90(327), 19–40 (2021)MathSciNetCrossRefMATH

12.

Kopteva, N., Meng, X.: Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions. SIAM J. Numer. Anal. 58(2), 1217–1238 (2020)MathSciNetCrossRefMATH

13.

Liao, H., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56(2), 1112–1133 (2018)MathSciNetCrossRefMATH

14.

Liao, H., McLean, W., Zhang, J.: A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218–237 (2019)MathSciNetCrossRefMATH

15.

Liao, H.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. Commun. Comput. Phys. 30(2), 567–601 (2021)MathSciNetCrossRefMATH

16.

Liao, H., Tang, T., Zhou, T.: A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations. J. Comput. Phys. 414, 109473 (2020)MathSciNetCrossRefMATH

17.

Liao, H.: An energy stable and maximum bound preserving scheme with variable time steps for time fractional allen-cahn equation. SIAM J. Sci. Comput. 43(5), A3503–A3526 (2021)MathSciNetCrossRefMATH

18.

Liao H. L.: Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations, to appear in J. Comput. Math. (2022)

19.

Liao, H., Zhang, Z.: Analysis of adaptive BDF2 scheme for diffusion equations. Math. Comput. 90(329), 1207–1226 (2021)MathSciNetCrossRefMATH

20.

Lv, C., Chuanju, X.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38(5), A2699–A2724 (2016)MathSciNetCrossRefMATH

21.

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

22.

Mustapha, K., Abdallah, B., Khaled, M., Furati, A.: A discontinuous Petrov-Galerkin method for time-fractional diffusion equations. SIAM J. Numer. Anal. 52(5), 2512–2529 (2014)MathSciNetCrossRefMATH

23.

Quan, C., Tang, T., Yang, J.: How to define dissipation-preserving energy for time-fractional phase-field equations. CSIAM Trans. Appl. Math. 1(3), 478–490 (2020)CrossRef

24.

Quan, C., Wu, Xu.: \({H^1}\)-stability of an L2-type method on general nonuniform meshes for subdiffusion equation, arXiv preprint arXiv:2205.06060 (2022)

25.

Stynes, M., O’Riordan, E.: and José Luis Gracia, 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)MathSciNetCrossRefMATH

26.

Tang, T., Haijun, Yu., Zhou, T.: On energy dissipation theory and numerical stability for time-fractional phase-field equations. SIAM J. Sci. Comput. 41(6), A3757–A3778 (2019)MathSciNetCrossRefMATH

Title: Global-in-Time -Stability of L2-1 Method on General Nonuniform Meshes for Subdiffusion Equation
Authors: Chaoyu Quan
Xu Wu
Publication date: 01-05-2023
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2023
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-023-02184-8

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