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Journal of Scientific Computing

Erschienen in:

10.05.2015

A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

verfasst von: Seakweng Vong, Pin Lyu, Zhibo Wang

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2016

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Abstract

In this paper, a compact finite difference scheme with global convergence order \(O\big (\tau ^{2-\alpha }+h^4\big )\) is derived for fractional sub-diffusion equations with the spatially variable coefficient subject to Neumann boundary conditions. The difficulty caused by the variable coefficient and the Neumann boundary conditions is overcome by subtle decomposition of the coefficient matrices. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. The theoretical results are supported by numerical experiments.

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Solomon, T.H., Weeks, E.R., Swinney, H.L.: Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow. Phys. Rev. Lett. 71, 3975–3979 (1993)CrossRef

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

Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATH

Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science and Technology, Amsterdam (2006)MATH

Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)CrossRefMathSciNetMATH

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

Chen, C., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740–1760 (2010)CrossRefMathSciNetMATH

Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)CrossRefMathSciNetMATH

Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)CrossRefMathSciNetMATH

10.

Gao, G., Sun, Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)CrossRefMathSciNetMATH

11.

Zhao, X., Sun, Z.: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 230, 6061–6074 (2011)CrossRefMathSciNetMATH

12.

Ren, J., Sun, Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456–467 (2013)CrossRefMathSciNetMATH

13.

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)CrossRefMathSciNetMATH

14.

Mohebbi, A., Abbaszadeh, M.: Compact finite difference scheme for the solution of time fractional advection–dispersion equation. Numer. Algorithm. 63, 431–452 (2013)CrossRefMathSciNetMATH

15.

Mohebbi, A., Abbaszadeh, M., Dehghan, M.: A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term. J. Comput. Phys. 240, 36–48 (2013)CrossRefMathSciNetMATH

16.

Abbaszadeh, M., Mohebbi, A.: A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term. Comput. Math. Appl. 66, 1345–1359 (2013)CrossRefMathSciNet

17.

Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)CrossRefMathSciNet

18.

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

19.

Zoppou, C., Knight, J.H.: Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions. Appl. Math. Model. 23, 667–685 (1999)CrossRefMATH

20.

Lai, M., Tseng, Y.: A fast iterative solver for the variable coefficient diffusion equation on a disk. J. Comput. Phys. 208, 196–205 (2005)CrossRefMathSciNetMATH

21.

Sun, Z.: An unconditionally stable and \(O(\tau ^2+h^4)\) order \(L_\infty \) convergence difference scheme for linear parabolic equation with variable coefficients. Numer. Methods Part Differ. Equ. 17, 619–631 (2001)CrossRefMATH

22.

Britt, S., Tsynkov, S., Turkel, E.: A compact fourth order scheme for the Helmholtz equation in polar coordinates. J. Sci. Comput. 45, 26–47 (2010)CrossRefMathSciNetMATH

23.

Medvinsky, M., Tsynkov, S., Turkel, E.: The method of difference potentials for the Helmholtz equation using compact high order schemes. J. Sci. Comput. 53, 150–193 (2012)CrossRefMathSciNetMATH

24.

Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model. 38, 3848–3859 (2014)CrossRefMathSciNet

25.

Vong, S., Wang, Z.: A high order compact finite difference scheme for time fractional Fokkee–Planck equations. Appl. Math. Lett. 43, 38–43 (2015)CrossRefMathSciNet

26.

Cui, M.: Compact exponential scheme for the time fractional convection–diffusion reaction equation with variable coefficients. J. Comput. Phys. 280, 143–163 (2015)CrossRefMathSciNet

27.

Vong, S., Wang, Z.: A high-order compact scheme for the nonlinear fractional Klein–Gordon equation. Numer. Methods Part Differ. Equ. (2014). doi:10.1002/num.21912

28.

Wang, Z., Vong, S.: A high-order exponential ADI scheme for two dimensional time fractional convection–diffusion equations. Comput. Math. Appl. 68, 185–196 (2014)CrossRefMathSciNet

29.

Vong, S., Wang, Z.: A compact difference scheme for a two dimensional fractional Klein–Gordon equation with Neumann boundary conditions. J. Comput. Phys. 274, 268–282 (2014)CrossRefMathSciNet

30.

Vong, S., Wang, Z.: High order difference schemes for a time fractional differential equation with Neumann boundary conditions. East Asian J. Appl. Math. 4, 222–241 (2014)MathSciNetMATH

31.

Cui, M.: Combined compact difference scheme for the time fractional convection–diffusion equation with variable coefficients. Appl. Math. Comput. 246, 464–473 (2014)CrossRefMathSciNet

32.

Sun, Z., Wu, X.: A fully discrete scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)CrossRefMathSciNetMATH

33.

Sun, Z.: Numerical Methods of Partial Differential Equations, 2nd edn. Science Press, Beijing (2012). (in Chinese)

Titel: A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions
verfasst von: Seakweng Vong
Pin Lyu
Zhibo Wang
Publikationsdatum: 10.05.2015
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0040-5

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