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Journal of Scientific Computing
Published in: Journal of Scientific Computing 1/2015

01-10-2015

Second-Order Stable Finite Difference Schemes for the Time-Fractional Diffusion-Wave Equation

Author: Fanhai Zeng

Published in: Journal of Scientific Computing | Issue 1/2015

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Abstract

We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton–Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection–reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones.
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Literature
1.
Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S.: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. (2014, in press)
2.
Cao, J.Y., Xu, C.J.: A high order schema for the numerical solution of the fractional ordinary differential equations. J. Comput. Phys. 238, 154–168 (2013)MathSciNetCrossRefMATH
3.
Chen, J., Liu, F., Anh, V., Shen, S., Liu, Q., Liao, C.: The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 219, 1737–1748 (2012)MathSciNetCrossRefMATH
4.
5.
Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)MathSciNetCrossRefMATH
6.
7.
Diethelm, K., Ford, N.J., Freed, A.D., Weilbeer, M.: Pitfalls in fast numerical solvers for fractional differential equations. J. Comput. Appl. Math. 186, 482–503 (2006)MathSciNetCrossRefMATH
8.
Ding, H.F., Li, C.P.: Numerical algorithms for the fractional diffusion-wave equation with reaction term. Abstr. Appl. Anal. 2013, 493406 (2013)MathSciNet
9.
Du, R., Cao, W.R., Sun, Z.Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34, 2998–3007 (2010)MathSciNetCrossRefMATH
10.
Hanygad, A.: Multidimensional solutions of time-fractional diffusion-wave equations. Proc. R. Soc. Lond. A 458, 933–957 (2002)CrossRef
11.
Huang, J., Tang, Y., Vázquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64, 707–720 (2013)MathSciNetCrossRefMATH
12.
Jafari, M.A., Aminataei, A.: An algorithm for solving multi-term diffusion-wave equations of fractional order. Comput. Math. Appl. 62, 1091–1097 (2011)MathSciNetCrossRefMATH
13.
Jafari, H., Momani, S.: Solving fractional diffusion and wave equations by modified homotopy perturbation method. Physics Letters A 370, 388–396 (2007)MathSciNetCrossRefMATH
14.
15.
Li, C.P., Zeng, F.H.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34, 149–179 (2013)MathSciNetCrossRefMATH
16.
Li, L.M., Xu, D., Luo, M.: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255, 471–485 (2013)MathSciNetCrossRef
17.
Lin, R., Liu, F.: Fractional high order methods for the nonlinear fractional ordinary differential equation. Nonlinear Anal. 66, 856–869 (2007)MathSciNetCrossRefMATH
18.
Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 9–25 (2013)
19.
20.
Lubich, C.: A stability analysis of convolution quadratures for Abel-Volterra integral equations. IMA J. Numer. Anal. 6, 87–101 (1986)MathSciNetCrossRefMATH
21.
Luchko, Y., Mainardi, F., Povstenko, Y.: Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation. Comput. Math. Appl. 66, 774–784 (2013)MathSciNetCrossRef
22.
23.
Mao, Z., Xiao, A.G., Yu, Z.G., Shi, L.: Sinc-Chebyshev collocation method for a class of fractional diffusion-wave equations. Sci. World J. 2014, 143983 (2014)
24.
McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)MathSciNetCrossRefMATH
25.
Metzler, R., Nonnenmacher, T.F.: Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation. Chem. Phys. 284, 67–90 (2002)CrossRef
26.
Murillo, J.Q., Yuste, S.B.: An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form. J. Comput. Nonlinear Dyn. 6, 021014 (2011)CrossRef
27.
Murillo, J.Q., Yuste, S.B.: A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations. Eur. Phys. J. Spec. Top. 222, 1987–1998 (2013)CrossRef
28.
Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491–515 (2013)MathSciNetCrossRefMATH
29.
Podlubny, I.: Fractional Differential Equations. Acdemic Press, San Dieg (1999)MATH
30.
Ren, J.C., Sun, Z.Z.: Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions. J. Sci. Comput. 56, 381–408 (2013)MathSciNetCrossRefMATH
31.
32.
33.
Sweilam, N.H., Khader, M.M., Adel, M.: On the stability analysis of weighted average finite difference methods for fractional wave equation. Fract. Differ. Calc. 2, 17–29 (2012)MathSciNetCrossRef
34.
Vázquez, L.: From Newton’s equation to fractional diffusion and wave equations. Adv. Differ. Equ. 2011, 169421 (2011)CrossRef
35.
Yang, J.Y., Huang, J.F., Liang, D.M., Tang, Y.F.: Numerical solution of fractional diffusion-wave equation based on fractional multistep method. Appl. Math. Model. 38, 3652–3661 (2014)MathSciNetCrossRef
36.
Zeng, F.H., Li, C.P., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)MathSciNetCrossRefMATH
37.
Zeng, F.H., Li, C.P., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. (2014, in press)
38.
Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)MathSciNetCrossRefMATH
Metadata
Title
Second-Order Stable Finite Difference Schemes for the Time-Fractional Diffusion-Wave Equation
Author
Fanhai Zeng
Publication date
01-10-2015
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-014-9966-2

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