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Journal of Scientific Computing
Published in: Journal of Scientific Computing 2/2017

11-02-2017

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

Authors: Xiaoli Li, Hongxing Rui

Published in: Journal of Scientific Computing | Issue 2/2017

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Abstract

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete \(L^{\infty }(L^2)\) and \(L^2(H^1)\) errors are \(O(\triangle t^{2-\alpha }+h^2+H^3)\). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.
previous article Optimized Overlapping Schwarz Waveform Relaxation for a Class of Time-Fractional Diffusion Problems
next article Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

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Literature
1.
Golmankhaneh, A.K., Baleanu, D.: Fractal calculus involving gauge function. Commun. Nonlinear Sci. Numer. Simul. 37, 125–130 (2016)MathSciNetCrossRef
2.
Khalili Golmankhaneh, A., Baleanu, D.: New derivatives on the fractal subset of real-line. Entropy 18(2), 1 (2016)CrossRef
3.
Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235(11), 3285–3290 (2011)MathSciNetCrossRefMATH
4.
Li, W., Da, X.: Finite central difference/finite element approximations for parabolic integro-differential equations. Computing 90(3–4), 89–111 (2010)MathSciNetCrossRefMATH
5.
Liu, Y., Du, Y., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem. Comput. Math. Appl. 70(4), 573–591 (2015)
6.
Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976–A3000 (2013)MathSciNetCrossRefMATH
7.
Zhang, N., Deng, W., Wu, Y.: Finite difference/element method for a two-dimensional modified fractional diffusion equation. Adv. Appl. Math. Mech 4, 496–518 (2012)MathSciNetCrossRefMATH
8.
Liu, Y., Fang, Z., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)MathSciNetMATH
9.
Liu, Y., Li, H., Gao, W., He, S., Fang, Z.: A new mixed element method for a class of time-fractional partial differential equations. Sci. World J. 2014, 141467 (2014). doi:10.​1155/​2014/​141467
10.
Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64(4), 707–720 (2013)MathSciNetCrossRefMATH
11.
Sousa, E.: Finite difference approximations for a fractional advection diffusion problem. J. Comput. Phys. 228(11), 4038–4054 (2009)MathSciNetCrossRefMATH
12.
Sousa, E.: A second order explicit finite difference method for the fractional advection diffusion equation. Comput. Math. Appl. 64(10), 3141–3152 (2012)MathSciNetCrossRefMATH
13.
14.
Vong, S., Wang, Z.: A high order compact finite difference scheme for time for fractional Fokker–Planck equations. Appl. Math. Lett. 43, 38–43 (2015)MathSciNetCrossRefMATH
15.
Liu, Z., Li, X.: A Crank–Nicolson difference scheme for the time variable fractional mobile-immobile advection-dispersion equation. J. Appl. Math. Comput. doi:10.​1007/​s12190-016-1079-7
16.
17.
Cheng, A., Wang, H., Wang, K.: A eulerian-lagrangian control volume method for solute transport with anomalous diffusion. Numer. Methods Partial Differ. Equ. 31(1), 253–267 (2015)MathSciNetCrossRefMATH
18.
Wei, L., He, Y.: Analysis of a fully discrete local discontinuous galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38(4), 1511–1522 (2014)MathSciNetCrossRef
19.
Lin, Y., Li, X., Xu, C.: Finite difference/spectral approximations for the fractional cable equation. Math. Comput. 80(275), 1369–1396 (2011)MathSciNetCrossRefMATH
20.
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)MathSciNetCrossRefMATH
21.
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl 1(2), 1–13 (2015)
22.
Atangana, A.: On the new fractional derivative and application to nonlinear fishers reaction-diffusion equation. Appl. Math. Comput. 273, 948–956 (2016)MathSciNet
23.
Atangana, A., Koca, I.: Chaos in a simple nonlinear system with atangana-baleanu derivatives with fractional order. Chaos, Solitons Fractals: Interdiscip. J. Nonlinear Sci., Nonequilib. Complex Phenom. 10, 447–454 (2016)
24.
Atangana, A., Baleanu, D.: Numerical solution of a kind of fractional parabolic equations via two difference schemes. Abstr. Appl. Anal. 2013, 828764 (2013). doi:10.​1155/​2013/​828764
25.
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)MathSciNetCrossRefMATH
26.
Bouzid, N., Merad, M., Baleanu, D.: On fractional Duffin–Kemmer–Petiau equation. Few-Body Syst. 57(4), 265–273 (2016)CrossRef
27.
Zhang, Y., Benson, D.A., Reeves, D.M.: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Resour. 32(4), 561–581 (2009)CrossRef
28.
Gao, G.H., Sun, H.W., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J. Comput. Phys. 280, 510–528 (2015)MathSciNetCrossRefMATH
29.
30.
Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64(10), 2990–3007 (2012)MathSciNetCrossRefMATH
31.
Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66(5), 693–701 (2013)MathSciNetCrossRefMATH
32.
Ashyralyev, A., Cakir, Z.: On the numerical solution of fractional parabolic partial differential equations with the dirichlet condition. Discrete Dyn. Nat. Soc. 2012, 696179 (2012). doi:10.​1155/​2012/​696179
33.
Ashyralyev, A., Cakir, Z.: Fdm for fractional parabolic equations with the neumann condition. Adv. Differ. Equ. 2013(1), 1–16 (2013)MathSciNetCrossRef
34.
35.
Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer Berlin Heidelberg (1977)
36.
Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J Numer Anal 34(2), 828–852 (1997)MathSciNetCrossRefMATH
37.
Rui, H., Pan, H.: A block-centered finite difference method for the darcy-forchheimer model. SIAM J. Numer. Anal. 50(5), 2612–2631 (2012)MathSciNetCrossRefMATH
38.
Li, X., Rui, H.: Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation. Int. J. Comput. Math. 89(1), 386–404 (2017)MathSciNetCrossRefMATH
39.
Li, X., Rui, H.: A two-grid block-centered finite difference method for nonlinear non-fickian flow model. Appl. Math. Comput. 281, 300–313 (2016)MathSciNet
40.
Rui, H., Pan, H.: Block-centered finite difference methods for parabolic equation with time-dependent coefficient. Jpn. J. Ind. Appl. Math. 30(3), 681–699 (2013)MathSciNetCrossRefMATH
41.
Rui, H., Wei, L.: A two-grid block-centered finite difference method for Darcy–Forchheimer flow in porous media. SIAM J. Numer. Anal. 53(4), 1941–1962 (2015)MathSciNetCrossRefMATH
42.
Liu, Z., Li, X.: A parallel cgs block-centered finite difference method for a nonlinear time-fractional parabolic equation. Comput. Methods Appl. Mech. Eng. 308, 330–348 (2016)MathSciNetCrossRef
43.
44.
Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35(2), 435–452 (1998)MathSciNetCrossRefMATH
45.
46.
Douglas Jr., J., Wang, J.: Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26(2–4), 121–133 (1989)MathSciNetCrossRefMATH
47.
Metadata
Title
A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation
Authors
Xiaoli Li
Hongxing Rui
Publication date
11-02-2017
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 2/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-017-0380-4

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