Banded Preconditioners for Two-Sided Space Variable-Order Fractional Diffusion Equations with a Nonlinear Source Term

Abstract

In this paper, we consider numerical methods for two-sided space variable-order fractional diffusion equations (VOFDEs) with a nonlinear source term. The implicit Euler (IE) method and a shifted Grünwald (SG) scheme are used to approximate the temporal derivative and the space variable-order (VO) fractional derivatives, respectively, which leads to an IE-SG scheme. Since the order of the VO derivatives depends on the space and the time variables, the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure. In light of the off-diagonal decay property of the coefficient matrices, we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems. The eigenvalue distribution and the condition number of the preconditioned matrices are studied. Numerical results show that the proposed banded preconditioners are efficient.

References

[1.] Axelsson O Iterative Solution Methods, 1994 Cambridge Cambridge University Press,

[2.] Axelsson, O., Kolotilina, L.: Montonicity and discretization error estimates. SIAM J. Numer. Anal. 27, 1591–1611 (1990)

[3.] Axelsson O, Kolotilina L. Diagonally compensated reduction and related preconditioning methods. Numer. Linear Algebra Appl., 1994, 1(2): 155-177,

[4.] Bai J, Feng X-C. Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Proc., 2007, 16: 2492-2502,

[5.] Bai Z-Z. Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput., 2000, 109: 273-285

[6.] Bai Z-Z. Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math., 2015, 283: 71-78,

[7.] Bai Z-Z, Duff IS, Wathen AJ. A class of incomplete orthogonal factorization methods. I: methods and theories. BIT Numer. Math., 2001, 41: 53-70,

[8.] Bai, Z.-Z., Lu, K.-Y.: On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations. BIT Numer. Math. 59, 1–33 (2019)

[9.] Bai Z-Z, Pan J-Y Matrix Analysis and Computations, 2021 Philadelphia SIAM,

[10.] Bai Z-Z, Yin J-F. Modified incomplete orthogonal factorization methods using Givens rotations. Computing, 2009, 86: 53-69,

[11.] Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)

[12.] Benson DA, Wheatcraft SW, Meerschaert MM. The fractional-order governing equation of Lévy motion. Water Resour. Res., 2000, 36: 1413-1423.

[13.] Diaz G, Coimbra C. Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dyn., 2009, 56(2): 145-157,

[14.] Du R, Alikhanov AA, Sun Z-Z. Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations. Comput. Math. Appl., 2020, 79: 2952-2972,

[15.] Hajipour, M., Jajarmi, A., Baleanu, D., Sun, H.: On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 69, 119–133 (2019)

[16.] Horn RA, Johnson CR Toptics in Matrix Analysis, 1994 Cambridge Academic Press

[17.] Ingman D, Suzdalnitsky J. Application of differential operator with servo-order function in model of viscoelastic deformation process. J. Eng. Mech., 2005, 131: 763-767.

[18.] Kobelev Y, Kobelev L, Klimontovich Y. Statistical physics of dynamic systems with variable memory. Dokl. Phys., 2003, 48: 285-289,

[19.] Kumar P, Chaudhary S. Analysis of fractional order control system with performance and stability. Int. J. Eng. Sci. Technol., 2017, 9: 408-416

[20.] Lei S-L, Sun H-W. A circulant preconditioner for fractional diffusion equations. J. Comput. Phys., 2013, 242: 715-725,

[21.] Lin, F.-R., Wang, Q.-Y., Jin, X.-Q.: Crank-Nicolson-weighted-shifted-Grünwald difference schemes for space Riesz variable-order fractional diffusion equations. Numer. Algorithms 87, 601–631 (2021)

[22.] Lin F-R, Yang S-W, Jin X-Q. Preconditioned iterative methods for fractional diffusion equation. J. Comput. Phys., 2014, 256: 109-117,

[23.] Lin R, Liu F, Anh V, Turner I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput., 2009, 212: 435-445

[24.] Lorenzo CF, Hartley TT. Variable-order and distributed order fractional operators. Nonlinear Dyn., 2002, 29: 57-98,

[25.] Lu X, Fang Z-W, Sun H-W. Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations. J. Appl. Math. Comput., 2020, 66: 673-700,

[26.] Miller KS, Ross B An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993 New York Wiley

[27.] Oldham KB, Spanier J The Fractional Calculus, 1974 New York Academic Press

[28.] Pan J-Y, Ke R-H, Ng MK, Sun H-W. Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput., 2014, 36: 2698-2719,

[29.] Pang, H.-K., Sun, H.-W.: A fast algorithm for the variable-order spatial fractional advection-diffusion equation. J. Sci. Comput. 87, 15 (2021)

[30.] Podlubny I Fractional Differential Equations, 1999 New York Cambridge University Press

[31.] Samko SG, Kilbas AA, Marichev OI Fractional Integerals and Derivatives: Theory and Applications, 1993 Amsterdam Gordon and Breach Science Publishers

[32.] Samko SG, Ross B. Integration and differentiation to a variable fractional order. Integral Transf. Spec. Funct., 1993, 1: 277-300,

[33.] Wang H, Wang K-X, Sircar T. A direct O ( N log 2 N ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O}({N}\log ^{2}{N})$$\end{document} finite difference method for fractional diffusion equations. J. Comput. Phys., 2010, 229: 8095-8104,

[34.] Wang, Q.-Y., She, Z.-H., Lao, C.-X., Lin, F.-R.: Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations. Numer. Algorithms 95, 859–895 (2024). https://doi.org/10.1007/s11075-023-01592-z

[35.] Zhao X, Sun Z-Z, Karniadakis GE. Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys., 2015, 293: 184-200,

[36.] Zhuang P, Liu F, Anh V, Turner I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal., 2009, 47: 1760-1781,