We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by β ∈[1, 2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pure advection (β = 1) and pure diffusion (β = 2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.
Keywords: fractional derivatives, local discontinuous Galerkin methods, stability, convergence, error estimates
@article{M2AN_2013__47_6_1845_0,
author = {Deng, W. H. and Hesthaven, J. S.},
title = {Local {Discontinuous} {Galerkin} methods for fractional diffusion equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1845--1864},
year = {2013},
publisher = {EDP Sciences},
volume = {47},
number = {6},
doi = {10.1051/m2an/2013091},
mrnumber = {3123379},
zbl = {1282.35400},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2013091/}
}
TY - JOUR AU - Deng, W. H. AU - Hesthaven, J. S. TI - Local Discontinuous Galerkin methods for fractional diffusion equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1845 EP - 1864 VL - 47 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2013091/ DO - 10.1051/m2an/2013091 LA - en ID - M2AN_2013__47_6_1845_0 ER -
%0 Journal Article %A Deng, W. H. %A Hesthaven, J. S. %T Local Discontinuous Galerkin methods for fractional diffusion equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1845-1864 %V 47 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2013091/ %R 10.1051/m2an/2013091 %G en %F M2AN_2013__47_6_1845_0
Deng, W. H.; Hesthaven, J. S. Local Discontinuous Galerkin methods for fractional diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1845-1864. doi: 10.1051/m2an/2013091
[1] , Sobolev Spaces. Academic Press, New York (1975). | Zbl | MR
[2] , Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E. 63 (2001) 046118.
[3] and , 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. | Zbl | MR
[4] and , An Introduction to Fractional Calculus. World Scientific, Singapore (2000). | Zbl | MR
[5] , , and , A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227 (2007) 886-897. | Zbl | MR
[6] and , The local discontinuous Galerkin method for time-dependent convection diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463. | Zbl | MR
[7] , , and , 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. | Zbl | MR
[8] , The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1975). | Zbl | MR
[9] , Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227 (2007) 1510-1522. | MR
[10] , Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47 (2008) 204-226. | MR
[11] and , Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Eqs. 22 (2005) 558-576. | Zbl | MR
[12] and , High-order nodal discontinuous Galerkin methods for Maxwell eigenvalue problem. Roy. Soc. London Ser. A 362 (2004) 493-524. | Zbl | MR
[13] and , Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer-Verlag, New York, USA (2008). | Zbl | MR
[14] and , 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. | Zbl | MR
[15] and , Remarks on fractional derivatives. Appl. Math. Comput. 187 (2007) 777-784. | Zbl | MR
[16] and , A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47 (2009) 2108-2131. | Zbl | MR
[17] and , Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007) 1533-1552. | Zbl | MR
[18] and , Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algorithms 52 (2009) 69-88. | Zbl | MR
[19] and , Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78 (2009) 1975-1995. | Zbl | MR
[20] and , Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algorithms 56 (2011) 159-184. | Zbl | MR
[21] and , Superconvergence of a discontinuous Galerkin method for the fractional diffusion and wave equation, arXiv:1206.2686v1 (2012). | Zbl | MR
[22] and , The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000) 1-77. | Zbl | MR
[23] and , A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813-823. | Zbl | MR
[24] and , A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769-791. | Zbl | MR
Cité par Sources :
