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Local Discontinuous Galerkin methods for fractional diffusionequations∗∗∗

Published online by Cambridge University Press:  07 October 2013

W.H. Deng and
J.S. Hesthaven
W.H. Deng
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China; Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.. dengwh@lzu.edu.cn
J.S. Hesthaven
Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.; Jan.Hesthaven@Brown.edu

Abstract

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We consider the development and analysis of local discontinuous Galerkin methods forfractional diffusion problems in one space dimension, characterized by having fractionalderivatives, parameterized by β ∈[1, 2]. After demonstrating that aclassic approach fails to deliver optimal order of convergence, we introduce a modifiedlocal numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pureadvection (β = 1) and pure diffusion (β = 2). In the twoclassic limits, known schemes are recovered. We discuss stability and present an erroranalysis for the space semi-discretized scheme, which is supported through a fewexamples.

Keywords

Fractional derivativeslocal discontinuous Galerkin methodsstabilityconvergenceerror estimates

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013, pp. 1845 - 1864
Copyright
© EDP Sciences, SMAI 2013

References

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Barkai, E., Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E. 63 (2001) 046118. Google ScholarPubMed
Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267–279. Google Scholar
P.L. Butzer and U. Westphal, An Introduction to Fractional Calculus. World Scientific, Singapore (2000).
Chen, C.-M., Liu, F., Turner, I. and Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227 (2007) 886–897. Google Scholar
Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. Google Scholar
Castillo, P., Cockburn, B., Schötzau, D. and Schwab, C., Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problem. Math. Comput. 71 (2001) 455–478. Google Scholar
P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1975).
Deng, W.H., Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227 (2007) 1510–1522. Google Scholar
Deng, W.H., Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47 (2008) 204–226. Google Scholar
Ervin, V.J. and Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Eqs. 22 (2005) 558–576. Google Scholar
Hesthaven, J.S. and Warburton, T., High-order nodal discontinuous Galerkin methods for Maxwell eigenvalue problem. Roy. Soc. London Ser. A 362 (2004) 493–524. Google ScholarPubMed
J.S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer-Verlag, New York, USA (2008).
Ji, X. and Tang, H., High-order accurate Runge-Kutta (Local) discontinuous Galerkin methods for one- and two-dimensional fractional diffusion equations. Numer. Math. Theor. Meth. Appl. 5 (2012) 333–358. Google Scholar
Li, C.P. and Deng, W.H., Remarks on fractional derivatives. Appl. Math. Comput. 187 (2007) 777–784. Google Scholar
Li, X.J. and Xu, C.J., A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47 (2009) 2108–2131. Google Scholar
Lin, Y.M. and Xu, C.J., Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007) 1533–1552. Google Scholar
McLean, W. and Mustapha, K., Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algorithms 52 (2009) 69–88. Google Scholar
Mustapha, K. and McLean, W., Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78 (2009) 1975–1995. Google Scholar
Mustapha, K. and McLean, W., Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algorithms 56 (2011) 159–184. Google Scholar
K. Mustapha and W. McLean, Superconvergence of a discontinuous Galerkin method for the fractional diffusion and wave equation, arXiv:1206.2686v1 (2012).
Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000) 1–77. Google Scholar
Tadjeran, C. and Meerschaert, M.M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813–823. Google Scholar
Yan, J. and Shu, C.-W., A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769–791. Google Scholar