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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1845-1864},
doi = {10.1051/m2an/2013091},
mrnumber = {3123379},
zbl = {1282.35400},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 1845-1864. doi : 10.1051/m2an/2013091. http://gdmltest.u-ga.fr/item/M2AN_2013__47_6_1845_0/
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