Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T10:11:50.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of an Implicit Fully Discrete Local Discontinuous Galerkin Method for the Time-Fractional Kdv Equation

Published online by Cambridge University Press:  29 May 2015

Leilei Wei ,
Yinnian He  and
Xindong Zhang
Leilei Wei
Affiliation:
College of Science, Henan University of Technology, Zhengzhou 450001, China
Yinnian He*
Affiliation:
Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Xindong Zhang
Affiliation:
College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China
*
*Corresponding author. Email: leileiwei09@gmail.com (L. Wei), heyn@mail.xjtu.edu.cn (Y. He), liaoyuan1126@163.com (X. Zhang)
Get access

Abstract

In this paper, we consider a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Korteweg-de Vries (KdV) equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditionally stable and convergent through analysis. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.

Keywords

65M6035K55Time-fractional partial differential equationsKdv equationlocal discontinuous Galerkin methodstabilityerror estimates
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 4 , August 2015 , pp. 510 - 527
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abdulaziz, O., Sami Bataineh, A. and Hashim, I., On convergence of homotopy analysis method and its modification for fractional modified KdV equations, J. Appl. Math. Comput., 33 (2010), pp. 61–81.Google Scholar
[2]Cheng, Y. and Shu, C.-W., A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comput., 77 (2007), pp. 699–730.Google Scholar
[3]Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), pp. 204–226.CrossRefGoogle Scholar
[4]Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52 (1989), pp. 411–435.Google Scholar
[5]Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for timedependent convection diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463.Google Scholar
[6]Cockburn, B., Kanschat, G., Perugia, I., and Schotzau, D., Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), pp. 264–285.Google Scholar
[7]Dong, B. and Shu, C.-W., Analysis of a local discontinuous Galerkin method for fourth-order timedependent problems, SIAM J. Numer. Anal., 47 (2009), pp. 3240–3268.Google Scholar
[8]Momani, S., An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul., 70 (2005), pp. 110–118.Google Scholar
[9]Murio, D., Implicite finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), pp. 1138–1145.Google Scholar
[10]Jiang, Y. and Ma, J., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), pp. 3285–3290.Google Scholar
[11]Liu, F.Anh, V.Turner, I. and Zhuang, P., Time fractional advection dispersion equation, J. Comput. Appl. Math., 13 (2003), pp. 233–245.Google Scholar
[12]Machado, J. T., Kiryakova, V. and Mainardi, F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 1140–1153.CrossRefGoogle Scholar
[13]Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.Google Scholar
[14]Metzler, R., Glöckle, W. G., and Nonnenmacher, T. F., Fractional model equation for anomalous diffusion, Phys. A, 211 (1994), pp. 13–24.CrossRefGoogle Scholar
[15]Momani, S. and Al-Khaled, K., Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (2005), pp. 1351–1365.Google Scholar
[16]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533–1552.Google Scholar
[17]Qiu, J., Liu, T. G. and Khoo, B. C., Runge-Kutta discontinuous Galerkin methods for compressible two-medium flow simulations: one-dimensional case, J. Comput. Phys., 222 (2007), pp. 353–373.CrossRefGoogle Scholar
[18]Schneider, W. R. and Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), pp. 134–144.CrossRefGoogle Scholar
[19]Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, 1993.Google Scholar
[20]Shao, L., Feng, X. and He, Y., The local discontinuous Galerkin finite element method for Burger’s equation, Math. Comput. Model., 54 (2011), pp. 2943–2954.Google Scholar
[21]Wei, L. L., He, Y., Zhang, X. and Wang, S., Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation, Finite Elem. Anal. Des., 59 (2012), pp. 28–34.Google Scholar
[22]Xia, Y., Xu, Y. and Shu, C.-W., Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, J. Comput. Phys., 227 (2007), pp. 472–491.CrossRefGoogle Scholar
[23]Xu, Y. and Shu, C.-W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), pp. 72–97.Google Scholar
[24]Xu, Y. and Shu, C.-W., Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Comput. Methods Appl. Mech. Eng., 196 (2007), pp. 3805–3822.Google Scholar
[25]Xu, Y. and Shu, C.-W., Local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal., 46 (2008), pp. 1998–2021.Google Scholar
[26]Yan, J. and Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40 (2002), pp. 769–791.CrossRefGoogle Scholar
[27]Yildirim, A., An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method, int. J. Nonlinear Sci. Numer. Simul., 10 (2009), pp. 445–450.Google Scholar
[28]Yildirim, A., He’s homtopy perturbation method for solving the space and time fractional telegraph equations, int. J. Comput. Math., 87 (2010), pp. 2998–3006.Google Scholar
[29]Yildirim, A., Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method, int. J. Numer. Methods Heat Fluid Flow., 20 (2010), pp. 186–200.CrossRefGoogle Scholar
[30]Zhang, Q. and Gao, F., A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation, J. Sci. Comput., 51 (2012), pp. 107–134.CrossRefGoogle Scholar
[31]Zhang, X. D., Tang, B. and He, Y. N., Homotopy analysis method for higher-order fractional integro-differential equations, Comput. Math. Appl., 62 (2011), pp. 3194–3203.Google Scholar
[32]Zhao, X. and Sun, Z. Z., A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), pp. 6061–6074.Google Scholar
[33]Zhang, Y. and Sun, Z. Z., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), pp. 8713–8728.Google Scholar