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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 898217, 11 pages

http://dx.doi.org/10.1155/2014/898217
Research Article

A Computational Study of an Implicit Local Discontinuous Galerkin Method for Time-Fractional Diffusion Equations

1Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China

2College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China

Received 15 February 2014; Accepted 8 July 2014; Published 19 August 2014

Academic Editor: Carlos Vazquez

Copyright © 2014 Leilei Wei and Xindong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose, analyze, and test a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully, we prove that our scheme is unconditionally stable and convergent. Finally, numerical examples are performed to illustrate the effectiveness and the accuracy of the method.

1. Introduction

Fractional calculus which is considered as the generalization of the integer order calculus attracts much attention recently its of their numerous applications in physics and engineering. They provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classic integral-order models, in which such effects are, in fact, neglected. Interest of some scholars has been shown in research on the problems involving the fractional order partial differential equations (PDEs) [119]. Machado et al. [20] introduced the recent history of fractional calculus; as for the detailed theory and applications of fractional integrals and derivatives, we can refer to [21, 22] and the references therein. Due to their numerous applications in the areas of physics and engineering, solving such equations and numerical schemes for fractional differential equations has been stimulated.

Fractional equations arise in continuous-time random walks, modeling of anomalous diffusive and subdiffusive systems, unification of diffusion and wave propagation phenomenon, and simplification of the results. There are only a few numerical works in the literature to solve fractional diffusion equations. Liu et al. [23] use a first-order finite difference scheme in both time and space directions for this equation, where some stability conditions are derived. In [14] Lin and Xu examine a practical finite difference/Legendre spectral method to solve the initial-boundary value time-fractional diffusion problem on a finite domain. In [11], Jiang and Ma use high-order finite element methods to solve the equation and prove an optimal convergence rate.

In this paper, we consider the following time-fractional diffusion equation: where is the order of the time-fractional derivatives. and are given smooth functions. We do not pay attention to boundary condition in this paper; hence, the solution is considered to be either periodic or compactly supported.

We define as the Caputo fractional derivatives of order [24, 25], here, is the Gamma function.

In the present paper, we propose a fully discrete local discontinuous Galerkin (LDG) finite element method for solving the time-fractional diffusion equation. Our fully discrete scheme is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully, we prove that our scheme is unconditionally stable and gives an error estimate.

What remains of this paper is organized as follows. We begin by introducing some basic notations and mathematical preliminaries which are required for establishing our results. In Section 3, we discuss the LDG scheme for the fractional equation (1), and we prove that the fully discrete scheme is unconditionally stable and convergent. Numerical experiments to illustrate the accuracy and capability of the method are given in Section 4. Finally, in Section 5, concluding remarks are provided.

2. Notations and Auxiliary Results

In this section, we introduce notations and definitions to be used later in the paper and also present some auxiliary results.

Given a spatial grid , define the mesh , for and the cell lengths , and .

We denote by and the values of at , from the right cell and from the left cell . is used to denote , that is, the jump of at cell interfaces.

We define the piecewise-polynomial space as the space of polynomials of the degree up to in each cell ; that is,

For error estimates, we will be using two projections in one dimension , denoted by ; that is, for each , and special projection ; that is, for each ,

For the two projections, the following inequality holds [2628]: where or . The positive constant , solely depending on , is independent of . denotes the set of boundary points of all elements .

In the present paper, we use to denote a positive constant which may have a different value in each occurrence. The usual notation of norms in Sobolev spaces will be used. Let the scalar inner product on be denoted by and the associated norm by . If , we drop .

3. Fully Discrete LDG Scheme

Let be the time meshsize, let be a positive integer, let be mesh point. First, we estimate the time-fractional derivatives at as follows [11, 14]: where , , is dependent on .

We rewrite (1) as a first-order system: and we give the weak form of (9) at as follows: where .

Let be the approximation of ; respectively, . We define a fully discrete local discontinuous Galerkin scheme as follows: find , such that, for all test functions ,

The “hat” terms in (11) in the cell boundary terms from integration by parts are the so-called “numerical fluxes,” which are single valued functions defined on the edges and should be designed based on different guiding principles for different PDEs to ensure stability. It turns out that we can take the simple choices such that

We remark that the choice for the fluxes (12) is not unique. In fact, the crucial part is taking and from opposite sides. We know the truncation error is from (8).

In order to simplify the notations and without loss of generality, we consider the case in its numerical analysis. Now, we consider the stability for the scheme (11), we have the following result.

Theorem 1. For periodic or compactly supported boundary conditions, the fully-discrete LDG scheme (11) is unconditionally stable, and the numerical solution satisfies

Proof. We will prove Theorem 1 by mathematical induction. When , scheme (11) is

Taking the test functions , we obtain Here,

If we take fluxes (12) and after some manual calculation, we can easily obtain .

From the fact that we can get

Now, suppose the following inequality holds We need to prove . Let and take the test functions in scheme (11); we can obtain

Taking fluxes (12), we can easily obtain . Then, the last inequality gives

This finishes the proof of the stability result.

Theorem 2. Let be the exact solution of the problem (1), which is sufficiently smooth with bounded derivatives. Let be the numerical solution of the fully discrete LDG scheme (11); then, there hold the following error estimates when : and when :

Proof. We denote Subtracting (11) from (10) and with fluxes (12), we can obtain the error equation Using (24), the error equation (25) can be written as Taking the test functions in (26), using the properties (4) and (6), then the following equality holds: that is, Based on the fact that , we can obtain For the sake of convenience, we denote (1) We start with the following estimate: When , (35) becomes Denoting , then we can obtain Next, we suppose the following inequality holds: Let in the inequality (35); we deduce

Notice the fact that we know that is,

Inequality (31) follows.

By some calculations and analyses, we know that increasingly tends to . For more details of the proof, we refer to [14]. So we can obtain

(2) The above estimate has no meaning when due to . So we must reconsider it for the case .

We suppose the following estimate holds:

By the similar techniques used in (1) and that in [14], we can obtain (40) easily. Here, we omitted the proof to save space. Then, we know that when ,

Thus, Theorem 2 follows by the triangle inequality and the interpolating property (7).

4. Numerical Examples

In this section, we offer some numerical examples to illustrate the accuracy and capability of the method. For this purpose, we calculate the numerical results of the exact solutions (for the cases where exact solutions are available). We mainly focus on the spatial accuracy, so a small time step is used such that errors stemming from the temporal approximation are negligible. With the aid of successive mesh refinements, we have verified that the results shown are numerically convergent.

Example 1. We consider time-fractional equation (1) in ; the corresponding forcing term is of the form Then, the exact solution is . The space and time step is , respectively. We check the spatial accuracy by fixing the time step sufficiently small to avoid contamination of the temporal error. From Tables 1, 2, 3, 4, 5, 6, 7, 8, and 9, we can see that the errors in -norm and -norm attain optimal order of accuracy for piecewise polynomials for . In Table 10, we show the errors in -norm and -norm attains order of accuracy for two values of and .

tab1
Table 1: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab2
Table 2: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab3
Table 3: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab4
Table 4: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab5
Table 5: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab6
Table 6: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab7
Table 7: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab8
Table 8: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab9
Table 9: Spatial accuracy test for the fractional order equation with the forcing term (42). , , .
tab10
Table 10: Temporal accuracy test for the problem (1) with the forcing term (42) when , , and .

5. Conclusion

In this paper, an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is presented for solving a class of time-fractional diffusion equation. Numerical examples show that the combination of the backward differentiation in time and local discontinuous Galerkin (LDG) finite element method in space leads to an approximation of order for smooth enough solution. The scheme can be extended to solve the two or higher dimensional case easily, and the theoretical results are also valid. The results show that the LDG method is a powerful and efficient technique in solving this class of problems with fractional derivatives.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Plan for Scientific Innovation Talent of Henan University of Technology (no. 2013CXRC12), the High-Level Personal Foundation of Henan University of Technology (no. 2013BS041), and the NSF of China, Tian Yuan Special Foundation (no. 11326200). This work is also supported by FSRPHEXJ (no. XJEDU2014S033).

References

  1. O. P. Agrawal, S. I. Muslih, and D. Baleanu, “Generalized variational calculus in terms of multi-parameters fractional derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4756–4767, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. A. Atangana and P. D. Vermeulen, “Analytical solutions of a space-time fractional derivative of groundwater flow equation,” Abstract and Applied Analysis, vol. 2014, Article ID 381753, 11 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. Baleanu, A. Bhrawy, D. Torres, and S. Salahshour, “Fractional and time-scales differential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 365250, 2 pages, 2014. View at Publisher · View at Google Scholar
  4. C. Chen, F. Liu, and K. Burrage, “Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 754–769, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications, vol. 48, no. 7-8, pp. 1017–1033, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. N. J. Ford, M. M. Rodrigues, and N. Vieira, “A numerical method for the fractional Schrödinger type equation of spatial dimension two,” Fractional Calculus and Applied Analysis, vol. 16, no. 2, pp. 454–468, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. P. Guo, C. Zeng, C. Li, and Y. Chen, “Numerics for the fractional Langevin equation driven by the fractional Brownian motion,” Fractional Calculus and Applied Analysis, vol. 16, no. 1, pp. 123–141, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. G. González-Parra, B. Chen-Charpentier, and A. J. Arenasd, “Polynomial chaos for random fractional order differential equations,” Applied Mathematics and Computation, vol. 226, pp. 123–130, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  10. C. Hu, B. Liu, and S. Xie, “Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments,” Applied Mathematics and Computation, vol. 222, pp. 72–81, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Y. Jiang and J. Ma, “High-order finite element methods for time-fractional partial differential equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 11, pp. 3285–3290, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. C. Li, M. Kostić, and M. Li, “On a class of time-fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 15, no. 4, pp. 639–668, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Y. Liu, P. Lu, and I. Szanto, “Numerical analysis for a fractional differential time-delay model of HIV infection of CD4+ T-cell proliferation under antiretroviral therapy,” Abstract and Applied Analysis, vol. 2014, Article ID 291614, 13 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533–1552, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. L. Wei, Y. He, X. Zhang, and S. Wang, “Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation,” Finite Elements in Analysis and Design, vol. 59, pp. 28–34, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. X. Zhao and Z. Z. Sun, “A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions,” Journal of Computational Physics, vol. 230, no. 15, pp. 6061–6074, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. Yildirim, “An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 4, pp. 445–450, 2009. View at Scopus
  18. A. Yildirim, “Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20, no. 2, pp. 186–200, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. X. Zhang, J. Liu, L. Wei, and C. Ma, “Finite element method for Grwünwald-Letnikov time-fractional partial differential equation,” Applicable Analysis, vol. 92, no. 10, pp. 2103–2114, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  22. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, 1993.
  23. F. Liu, S. Shen, V. Anh, and I. Turner, “Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation,” The ANZIAM Journal, vol. 46, pp. C488–C504, 2004/05. View at MathSciNet
  24. A. Yıldırım, “He's homotopy perturbation method for solving the space- and time-fractional telegraph equations,” International Journal of Computer Mathematics, vol. 87, no. 13, pp. 2998–3006, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, New York, NY, USA, 1999.
  26. B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, “Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids,” SIAM Journal on Numerical Analysis, vol. 39, no. 1, pp. 264–285, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. Y. Xu and C.-W. Shu, “Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 37–40, pp. 3805–3822, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. Y. Xu and C. Shu, “A local discontinuous Galerkin method for the Camassa-Holm equation,” SIAM Journal on Numerical Analysis, vol. 46, no. 4, pp. 1998–2021, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus