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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 1/2019

21.09.2018

On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations

verfasst von: Zhong-Zhi Bai, Kang-Ya Lu

Erschienen in: BIT Numerical Mathematics | Ausgabe 1/2019

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Abstract

For solving time-dependent one-dimensional spatial-fractional diffusion equations of variable coefficients, we establish a banded M-splitting iteration method applicable to compute approximate solutions for the corresponding discrete linear systems resulting from certain finite difference schemes at every temporal level, and demonstrate its asymptotic convergence without imposing any extra condition. Also, we provide a multistep variant for the banded M-splitting iteration method, and prove that the computed solutions of the discrete linear systems by employing this iteration method converge to the exact solutions of the spatial fractional diffusion equations. Numerical experiments show the accuracy and efficiency of the multistep banded M-splitting iteration method.
Nächster Artikel Multivariate data fitting with error control

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Literatur
1.
Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)CrossRefMATHMathSciNet
2.
Bai, Z.-Z., Lu, K.-Y., Pan, J.-Y.: Diagonal and Toeplitz splitting iteration methods for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations. Numer. Linear Algebra Appl. 24(e2093), 1–15 (2017)MATHMathSciNet
3.
Bai, Z.-Z., Wang, D.-R.: Generalized matrix multisplitting relaxation methods and their convergence. Numer. Math. J. Chin. Univ. (Engl. Ser.) 2, 87–100 (1993)MATHMathSciNet
4.
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences (Revised Reprint of the 1979 Original). SIAM, Philadelphia (1994)
5.
Breiten, T., Simoncini, V., Stoll, M.: Low-rank solvers for fractional differential equations. Electron. Trans. Numer. Anal. 45, 107–132 (2016)MATHMathSciNet
6.
Carreras, B.A., Lynch, V.E., Zaslavsky, G.M.: Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas 8, 5096–5103 (2001)CrossRef
7.
Deng, W.-H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008/2009)
8.
Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)CrossRefMATHMathSciNet
9.
Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)CrossRefMATHMathSciNet
10.
Hu, J.-G.: The estimate of \(\Vert M^{-1}N\Vert _{\infty }\) and the optimally scaled matrix. J. Comput. Math. 2, 122–129 (1984)MathSciNet
11.
Kirchner, J.W., Feng, X.-H., Neal, C.: Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403, 524–527 (2000)CrossRef
12.
Kreer, M., Kızılersü, A., Thomas, A.W.: Fractional Poisson processes and their representation by infinite systems of ordinary differential equations. Stat. Probab. Lett. 84, 27–32 (2014)CrossRefMATHMathSciNet
13.
Lei, S.-L., Sun, H.-W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)CrossRefMATHMathSciNet
14.
Li, X.-J., Xu, C.-J.: A space–time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)CrossRefMATHMathSciNet
15.
Li, X.-J., Xu, C.-J.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8, 1016–1051 (2010)MATHMathSciNet
16.
Lin, F.-R., Yang, S.-W., Jin, X.-Q.: Preconditioned iterative methods for fractional diffusion equation. J. Comput. Phys. 256, 109–117 (2014)CrossRefMATHMathSciNet
17.
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Danbury (2006)
18.
Meerschaert, M.M., Scheffler, H.-P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249–261 (2006)CrossRefMATHMathSciNet
19.
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)CrossRefMATHMathSciNet
20.
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)CrossRefMATHMathSciNet
21.
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)CrossRefMATHMathSciNet
22.
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (2000)CrossRefMATH
23.
Pan, J.-Y., Ke, R.-H., Ng, M.K., Sun, H.-W.: Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput. 36, A2698–A2719 (2014)CrossRefMATHMathSciNet
24.
Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)MATH
25.
Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Phys. A Stat. Mech. Appl. 314, 749–755 (2002)CrossRefMATH
26.
Razminia, K., Razminia, A., Baleanu, D.: Investigation of the fractional diffusion equation based on generalized integral quadrature technique. Appl. Math. Model. 39, 86–98 (2015)CrossRefMATHMathSciNet
27.
Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distributions in financial markets. Eur. Phys. J. B Condens. Matter Phys. 27, 273–275 (2002)MathSciNet
28.
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, Yverdon (1993)MATH
29.
30.
Upadhyay, R.K., Mondal, A.: Dynamics of fractional order modified Morris–Lecar neural model. Netw. Biol. 5, 113–136 (2015)
31.
Varga, R.S.: Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs (1962)MATH
32.
33.
Wang, H., Wang, K.-X., Sircar, T.: A direct \({{\cal{O}}}(N\log ^2N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)CrossRefMATHMathSciNet
34.
Metadaten
Titel
On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations
verfasst von
Zhong-Zhi Bai
Kang-Ya Lu
Publikationsdatum
21.09.2018
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 1/2019
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-018-0727-8

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