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Journal of Scientific Computing

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01-07-2021

A Preconditioning Technique for an All-at-once System from Volterra Subdiffusion Equations with Graded Time Steps

Authors: Yong-Liang Zhao, Xian-Ming Gu, Alexander Ostermann

Published in: Journal of Scientific Computing | Issue 1/2021

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Abstract

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well. The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near \(t = 0\). In this paper, we propose a modification. We first split the time interval [0, T] into \([0, T_0]\) and \([T_0, T]\), where \(T_0\) (\(0< T_0 < T\)) is reasonably small. Then, the graded L1 scheme is applied in \([0, T_0]\), while the uniform one is used in \([T_0, T]\). Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.

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Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today 55, 48–54 (2002)

Metzler, R., Schick, W., Kilian, H.-G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)

He, J.-H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Meth. Appl. Mech. Eng. 167, 57–68 (1998)MathSciNetMATH

del-Castillo-Negrete, D., Carreras, B., Lynch, V.: Fractional diffusion in plasma turbulence. Phys. Plasmas 11, 3854–3864 (2004)

Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A 278, 107–125 (2000)MathSciNetMATH

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

Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J. Comput. Phys. 228, 3137–3153 (2009)MathSciNetMATH

Pagnini, G., Paradisi, P.: A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 19, 408–440 (2016)MathSciNetMATH

Podlubny, I.: Fractional Differential Equations, vol. 198. Academic Press, San Diego, CA (1998)MATH

10.

Lin, X.-L., Ng, M.K., Sun, H.-W.: Crank-Nicolson alternative direction implicit method for space-fractional diffusion equations with nonseparable coefficients. SIAM J. Numer. Anal. 57, 997–1019 (2019)MathSciNetMATH

11.

Lei, S.-L., Wang, W., Chen, X., Ding, D.: A fast preconditioned penalty method for American options pricing under regime-switching tempered fractional diffusion models. J. Sci. Comput. 75, 1633–1655 (2018)MathSciNetMATH

12.

Shen, J., Li, C., Sun, Z.-Z.: An H2N2 interpolation for Caputo derivative with order in (1,2) and its application to time-fractional wave equations in more than one space dimension. J. Sci. Comput. 83, 38 (2020). https://doi.org/10.1007/s10915-020-01219-8MathSciNetCrossRefMATH

13.

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

14.

Cao, J., Song, G., Wang, J., Shi, Q., Sun, S.: 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 (2019)MathSciNetMATH

15.

Gu, X.-M., Wu, S.-L.: A parallel-in-time iterative algorithm for Volterra partial integral-differential problems with weakly singular kernel. J. Comput. Phys. 417, 109576 (2020). https://doi.org/10.1016/j.jcp.2020.109576MathSciNetCrossRefMATH

16.

Zhao, Y.-L., Zhu, P.-Y., Gu, X.-M., Zhao, X.-L., Jian, H.-Y.: A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation. J. Sci. Comput. 83, 10 (2020). https://doi.org/10.1007/s10915-020-01193-1MathSciNetCrossRefMATH

17.

Li, M., Zhao, Y.-L.: A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator. Appl. Math. Comput. 338, 758–773 (2018)MathSciNetMATH

18.

Bouchaud, J.-P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)MathSciNet

19.

Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetMATH

20.

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

21.

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

22.

Zhang, Y.-N., Sun, Z.-Z., Liao, H.-L.: Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265, 195–210 (2014)MathSciNetMATH

23.

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

24.

Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2016)MathSciNetMATH

25.

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

26.

Hu, X., Rodrigo, C., Gaspar, F.J.: Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation. J. Comput. Phys. 416, 109540 (2020). https://doi.org/10.1016/j.jcp.2020.109540

27.

Mustapha, K., AlMutawa, J.: A finite difference method for an anomalous sub-diffusion equation, theory and applications. Numer. Algorithms 61, 525–543 (2012)MathSciNetMATH

28.

Mustapha, K.: An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements. IMA J. Numer. Anal. 31, 719–739 (2011)MathSciNetMATH

29.

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, 1057–1079 (2017)MathSciNetMATH

30.

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

31.

Lubich, C., Sloan, I., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65, 1–17 (1996)MathSciNetMATH

32.

Zeng, F., Zhang, Z., Karniadakis, G.E.: Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput. Meth. Appl. Mech. Eng. 327, 478–502 (2017)MathSciNetMATH

33.

Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56, 210–227 (2018)MathSciNetMATH

34.

Jin, B., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Meth. Appl. Mech. Eng. 346, 332–358 (2019)MathSciNetMATH

35.

Wang, Y., Yan, Y., Yan, Y., Pani, A.K.: Higher order time stepping methods for subdiffusion problems based on weighted and shifted Grünwald-Letnikov formulae with nonsmooth data. J. Sci. Comput. 83, 40 (2020). https://doi.org/10.1007/s10915-020-01223-yCrossRefMATH

36.

Kwon, K., Sheen, D.: A parallel method for the numerical solution of integro-differential equation with positive memory. Comput. Meth. Appl. Mech. Eng. 192, 4641–4658 (2003)MathSciNetMATH

37.

McLean, W., Thomée, V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation. IMA J. Numer. Anal. 30, 208–230 (2010)MathSciNetMATH

38.

Li, X., Tang, T., Xu, C.: Parallel in time algorithm with spectral-subdomain enhancement for Volterra integral equations. SIAM J. Numer. Anal. 51, 1735–1756 (2013)MathSciNetMATH

39.

Wu, S.-L., Zhou, T.: Parareal algorithms with local time-integrators for time fractional differential equations. J. Comput. Phys. 358, 135–149 (2018)MathSciNetMATH

40.

Fu, H., Wang, H.: A preconditioned fast parareal finite difference method for space-time fractional partial differential equation. J. Sci. Comput. 78, 1724–1743 (2019)MathSciNetMATH

41.

Ke, R., Ng, M.K., Sun, H.-W.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303, 203–211 (2015)MathSciNetMATH

42.

Huang, Y.-C., Lei, S.-L.: A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations. Numer. Algorithms 76, 605–616 (2017)MathSciNetMATH

43.

Lu, X., Pang, H.-K., Sun, H.-W.: Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub-diffusion equations. Numer. Linear Algebr. Appl. 22, 866–882 (2015)MathSciNetMATH

44.

Lu, X., Pang, H.-K., Sun, H.-W., Vong, S.-W.: Approximate inversion method for time-fractional subdiffusion equations. Numer. Linear Algebr. Appl. 25, e2132 (2018). https://doi.org/10.1002/nla.2132MathSciNetCrossRefMATH

45.

Bertaccini, D., Durastante, F.: Limited memory block preconditioners for fast solution of fractional partial differential equations. J. Sci. Comput. 77, 950–970 (2018)MathSciNetMATH

46.

Bertaccini, D., Durastante, F.: Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation. Appl. Math. Lett. 95, 92–97 (2019)MathSciNetMATH

47.

Bertaccini, D., Durastante, F.: Solving mixed classical and fractional partial differential equations using the short-memory principle and approximate inverses. Numer. Algorithms 74, 1061–1082 (2017)MathSciNetMATH

48.

Van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13, 631–644 (1992)MathSciNetMATH

49.

Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia, PA (2003)MATH

50.

Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21, 1969–1972 (2000)MathSciNetMATH

51.

Moroney, T., Yang, Q.: Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners. J. Comput. Phy. 246, 304–317 (2013)MathSciNetMATH

52.

Gu, X.-M., Zhao, Y.-L., Zhao, X.-L., Carpentieri, B., Huang, Y.-Y.: A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations. Numer. Math. Theor. Meth. Appl. (2021). https://doi.org/10.4208/nmtma.OA-2020-0020

53.

Liao, H.-L., Yan, Y., Zhang, J.: Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80, 1–25 (2019)MathSciNetMATH

54.

Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, PA (1997)MATH

Title: A Preconditioning Technique for an All-at-once System from Volterra Subdiffusion Equations with Graded Time Steps
Authors: Yong-Liang Zhao
Xian-Ming Gu
Alexander Ostermann
Publication date: 01-07-2021
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2021
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-021-01527-7

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